Graphical Discreteness, Coxeter Doublings and Generalized Polygons

1. Introduction

Graphical discreteness is not preserved by quasi-isometry in general [Corollary 1.3] [16]. The main result concerns a natural family of quasi-isometry classes on each of which it is nevertheless constant. If $\mathsf{\Gamma }$ is a finite thick generalized m-gon with $m\ge 3$ and $\mathsf{\Lambda }$ is any finitely generated group quasi-isometric to the right-angled Coxeter group $W_{\mathsf{\Gamma }}$, then

$$\mathsf{\Lambda }\mathrm{is}\mathrm{graphically}\mathrm{discrete}⟺\mathsf{\Gamma }\mathrm{is}\mathrm{nonflexible}.$$

Thus graphical discreteness is constant among finitely generated groups quasi-isometric to $W_{\mathsf{\Gamma }}$, even though it is not a quasi-isometry invariant in general. This is Theorem D below.

Graphical discreteness was introduced by Margolis, Shepherd, Stark and Woodhouse as a rigidity condition governing all geometric graph actions of a finitely generated group [§3.1] [16]. A finitely generated group $\mathsf{\Lambda }$ is graphically discrete if $Aut\left(X\right)$ is compact-by-discrete for every connected locally finite graph X on which $\mathsf{\Lambda }$ acts properly and cocompactly [Proposition 3.6(3)] [16]. The word “compact” is essential. A nondiscrete automorphism group can still be compact-by-discrete, so nondiscreteness of $Aut\left(X\right)$ for a single geometric graph action is not by itself an obstruction.

For the special case $\mathsf{\Lambda }=W_{\mathsf{\Gamma }}$, the nonflexible implication can already be recovered from the Davis-complex rigidity results of Haglund–Paulin [Théorème 5.12] [13], the right-angled finite-index formulation in [Remark 4.12] [19], Xie’s Fuchsian-building rigidity theorem [21], recalled in [Theorem 2.4] [3], the Bounds–Xie realization [Theorem 4.5] [3], and the finite-index quasi-isometry criterion [Theorem 3.27] [16]. Theorem 1.11(ii) of [19] also gives a short route to the nonflexible implication for arbitrary $\mathsf{\Lambda }$. Every such $\mathsf{\Lambda }$ is abstractly commensurable with $W_{\mathsf{\Gamma }}$. If $H\le W_{\mathsf{\Gamma }}$ has finite index, inclusion is a quasi-isometry and induces $QI\left(H\right)\cong QI\left(W_{\mathsf{\Gamma }}\right)$. Under this isomorphism, the left-multiplication image of H has finite index in the left-multiplication image of $W_{\mathsf{\Gamma }}$. The left-multiplication image of $W_{\mathsf{\Gamma }}$ has finite index in $QI\left(W_{\mathsf{\Gamma }}\right)$, as verified explicitly in the proof of Theorem 8. Moreover, H is non-elementary hyperbolic and hence tame, so [Theorem 3.27] [16] implies that H is graphically discrete. Let ${\mathsf{\Lambda }}_0\le \mathsf{\Lambda }$ and $H\le W_{\mathsf{\Gamma }}$ be isomorphic finite-index subgroups supplied by commensurability. Then ${\mathsf{\Lambda }}_0$ is graphically discrete, and so is $\mathsf{\Lambda }$, since graphical discreteness passes from a finite-index subgroup to the ambient group directly from the definition. This route cannot recover the flexible implication, since graphical discreteness need not pass to finite-index subgroups [Corollary 1.3] [16]. For that direction, [Corollary 1.2] [3] and the metric-strata lemma supply a common Coxeter Cayley-graph witness throughout the quasi-isometry class.

The contributions beyond these predecessor results are concentrated in the interfaces where graphical discreteness is stronger than ordinary rigidity. The flexible implication in Theorem D is proved uniformly for every finitely generated group in the quasi-isometry class. Theorem A upgrades the standard Coxeter Cayley-graph criterion from discreteness to compact-by-discreteness. Theorem B turns one-vertex doublings into additional finite-index witnesses. The later generalized-polygon and graph-product results make these criteria explicit in finite-geometric and building-theoretic families.

The remaining results give obstructions and structural criteria for Coxeter groups and graph products of finite groups, with applications to finite geometry. To state Theorem A, recall that the defining diagram of a Coxeter system $(W,S)$ has vertex set S, where distinct $s,t\in S$ are joined by a labelled edge exactly when $m_{st}<\infty$. It is flexible if there are $s\in S$ and a nontrivial label-preserving diagram automorphism $\varphi$ that fixes s and every vertex joined to s [p. 1] [20].

Theorem A.

Let $(W,S)$ be a Coxeter system with S finite. For the standard undirected Cayley graph $\mathcal{C}(W,S)$, the following conditions are equivalent.

1.

$Aut\left(\mathcal{C}\right(W,S\left)\right)$ is compact-by-discrete.

2.

$Aut\left(\mathcal{C}\right(W,S\left)\right)$ is discrete.

3.

the defining diagram of $(W,S)$ is nonflexible.

Consequently, a Coxeter group admitting a flexible Coxeter system is not graphically discrete.

Haglund–Paulin’s restriction isomorphism and rigid-case vertex-stabilizer calculation give the nonflexible direction for finite-rank Coxeter systems [Corollaire 5.8 and Théorème 5.12] [13]. Their flexible-side construction assumed word hyperbolicity. White later introduced a first-s Cayley-graph automorphism for flexible diagrams [Definition 21 and Proposition 26] [20]. Berlai and Ferov independently established the finite-rank nondiscreteness criterion and, more generally, characterized uncountable vertex stabilizers in the standard Coxeter Cayley graph [Corollary B and Theorem A] [1]. To rule out compact-by-discreteness, the first-s construction is verified directly, and conjugates of the resulting automorphism produce arbitrarily large finite vertex-stabilizer orbits. The stabilizer-orbit criterion of Margolis–Shepherd–Stark–Woodhouse [Theorem 3.10(1),(2)] [16] then rules out compact-by-discreteness, not merely discreteness.

Theorem A uses the standard Cayley graph of the Coxeter system. A different geometric witness arises from an index-two Coxeter subgroup. For a finite simplicial graph $\mathsf{\Gamma }$, write $V\mathsf{\Gamma }$ and $E\mathsf{\Gamma }$ for its vertex and edge sets. For $v\in V\mathsf{\Gamma }$, the Doubling Lemma [Lemma 2.7] [6] uses the graph $D_v\mathsf{\Gamma }$ formed by gluing two copies of $\mathsf{\Gamma }\setminus \left\{v\right\}$ along their copies of ${lk}_{\mathsf{\Gamma }}\left(v\right)$, and realizes $W_{D_v\mathsf{\Gamma }}$ as an index-two subgroup of $W_{\mathsf{\Gamma }}$.

Theorem B.

If $D_v\mathsf{\Gamma }$ is flexible for some $v\in V\mathsf{\Gamma }$, then the right-angled Coxeter group $W_{\mathsf{\Gamma }}$ is not graphically discrete.

Conjugation by v acts on the standard generating set of $W_{D_v\mathsf{\Gamma }}$ by the deck involution of $D_v\mathsf{\Gamma }$. Hence $W_{\mathsf{\Gamma }}$ acts geometrically on the standard Cayley graph of the doubled subgroup. If the doubled diagram is flexible, Theorem A shows that the full automorphism group of this graph is not compact-by-discrete. This argument also recovers the obstruction from proper partial conjugations.

Corollary C.

If $\mathsf{\Gamma }\setminus {st}_{\mathsf{\Gamma }}\left(v\right)$ has at least two connected components for some $v\in V\mathsf{\Gamma }$, then $W_{\mathsf{\Gamma }}$ is not graphically discrete. In particular, the existence of a proper partial conjugation obstructs graphical discreteness.

The first assertion of Corollary C is proved below as Corollary 13. The partial-conjugation formulation follows immediately afterward. Corollary 14, which applies when a link vertex misses a component of the complement of the relevant closed star, is called the link-gap criterion. Corollary 15, which applies when distinct vertices $s,v$ satisfy $st\left(s\right)\subseteq st\left(v\right)$ and $st\left(v\right)\ne V\mathsf{\Gamma }$, is called the dominated-star criterion. The doubling obstruction is strictly stronger than these two criteria. Corollary 16 constructs an infinite family ${\mathsf{\Theta }}_m$, $m\ge 2$, of pairwise nonisomorphic asymmetric graphs without dominated vertices for which the corresponding groups are one-ended, hyperbolic and not graphically discrete, although the link-gap criterion fails at every vertex.

The generalized-polygon theorem announced at the outset uses the Fuchsian-building metric of Bounds and Xie. They equip the Davis complex of $W_{\mathsf{\Gamma }}$ with a regular piecewise hyperbolic metric that makes it a Fuchsian building [Theorem 4.5] [3]. The classification obtained here is as follows.

Theorem D.

Let Γ be a finite thick generalized m-gon, where $m\ge 3$, and let Λ be any finitely generated group quasi-isometric to $W_{\mathsf{\Gamma }}$. Then

$$\mathsf{\Lambda }\mathit{is}\mathit{graphically}\mathit{discrete}⟺\mathsf{\Gamma }\mathit{is}\mathit{nonflexible}.$$

Thus graphical discreteness, although not a quasi-isometry invariant in general [Corollary 1.3] [16], is constant on each of these quasi-isometry classes.

By the Feit–Higman theorem, the finite thick case can occur only for $m\in \{3,4,6,8\}$ [9]. The uniform formulation $m\ge 3$ is retained in Theorem D.

Theorem D does not assert the existence of a nonflexible finite thick generalized m-gon with $m\ge 3$. The existence question is posed in Section 4. Proposition 11 uses standard root-group and elation actions to exclude two broad families. Every finite thick Moufang generalized polygon is flexible. The same holds for every finite thick elation generalized quadrangle and its dual. For a finite projective plane with nonflexible incidence graph, the full automorphism group of each one-vertex double is described explicitly, and nonflexibility passes to every double. Proposition 18 gives the precise projective-plane criterion. The incidence graph is nonflexible if and only if the plane has no nontrivial perspectivity. Baer’s involution theorem, as stated in [p. 878] [5], gives a further restriction. At nonsquare order, the collineation group of such a plane has odd order and is therefore solvable by the Feit–Thompson theorem [Chapter I, §1, p. 775] [10]. Thus a finite perspectivity-free projective plane would give a graphically discrete hyperbolic right-angled Coxeter group whose defining graph and all one-vertex doubles are nonflexible.

For a finite simplicial graph $\mathsf{\Gamma }$ with vertex set S and a family ${\left(G_s\right)}_{s\in S}$ of nontrivial finite groups, let $G_{\mathsf{\Gamma }}$ denote the corresponding graph product [Chapter 3] [12]. Its standard chamber graph ${\mathcal{C}}_{\mathsf{\Gamma }}$ has vertex set $G_{\mathsf{\Gamma }}$, with distinct vertices $g,h$ adjacent when $g^{-1}h\in G_s\setminus \left\{1\right\}$ for some $s\in S$ [Definitions 2.6 and 3.1] [19]. Put

$$T=\{s\in S:|G_s|>2\},{\mathsf{\Gamma }}_0=\mathsf{\Gamma }[S\setminus T].$$

Here $\mathsf{\Gamma }\left[A\right]$ denotes the subgraph induced by A. By the definition of flexibility, a graph with empty vertex set is nonflexible. A vertex is universal if it is adjacent to every other vertex. Kubena and Thomas determined the discreteness dichotomy for automorphism groups of regular right-angled buildings [Theorem 1] [15]. Theorem E gives the compact-by-discrete classification for the standard chamber graph.

Theorem E.

The following conditions are equivalent.

1.

$Aut\left({\mathcal{C}}_{\mathsf{\Gamma }}\right)$ is compact-by-discrete.

2.

$Aut\left({\mathcal{C}}_{\mathsf{\Gamma }}\right)$ is discrete.

3.

every vertex of T is universal in Γ and ${\mathsf{\Gamma }}_0$ is nonflexible.

In particular, if $G_{\mathsf{\Gamma }}$ is graphically discrete, then every vertex of T is universal in Γ and ${\mathsf{\Gamma }}_0$ is nonflexible.

Finally, Theorem A applies to a family of hyperbolic right-angled Coxeter groups with Menger curve boundary. Let $O_n=KG(2n-1,n-1)$ be the n-th odd graph, where $KG(N,r)$ denotes the Kneser graph whose vertices are the r-element subsets of an N-element set and whose edges join disjoint subsets [p. 5] [2].

Corollary F.

For every $n\ge 3$, the right-angled Coxeter group $W_{O_n}$ is hyperbolic, has Gromov boundary homeomorphic to the Menger curve, and is not graphically discrete.

Theorems A, B, D and E are proved below as Theorem 7, Theorem 12, Theorem 8 and Theorem 23, respectively. Corollary F is proved below as Corollary 26. Corollary C is located above.

2. Preliminaries

An isometric action is called geometric if it is proper, cocompact and continuous in the sense of [Definition 2.22] [16]. Finitely generated groups carry the discrete topology [Remark 2.23] [16]. Unless stated otherwise, geometric graph actions are on connected locally finite graphs. Defining graphs for graph products and right-angled Coxeter groups are finite simplicial graphs and may be disconnected. Automorphism groups of locally finite graphs carry the topology of pointwise convergence on vertices. For a proper metric space Y, the group $Isom\left(Y\right)$ is equipped here with the compact-open topology. With this topology it acts properly and continuously on Y [§2.4, immediately after Remark 2.23] [16].

Definition 1 ([§3.1 and Proposition 3.6(3)] [16]). A locally compact group H is compact-by-discrete if it has a compact normal subgroup K such that $H/K$ is discrete. Using the equivalent graph action formulation, a finitely generated group $\mathsf{\Lambda }$ is graphically discrete if $Aut\left(X\right)$ is compact-by-discrete for every geometric action $\mathsf{\Lambda }\curvearrowright X$ on a connected locally finite graph.

Margolis–Shepherd–Stark–Woodhouse give the following compact-by-discrete criterion.

Theorem 2 ([Theorem 3.10(1),(2)] [16]). Let X be a connected locally finite graph, and let $H\le Aut\left(X\right)$ be a closed subgroup acting cocompactly on X. Then H is compact-by-discrete if and only if the cardinalities

$$|H_x·y|,x,y\in V\left(X\right),$$

are uniformly bounded.

Only the contrapositive will be used. In this setting, unbounded stabilizer orbits rule out compact-by-discreteness.

A second criterion comes from quasi-isometries. A finitely generated group is tame in the sense of [Definition 2.28] [16] if quasi-isometries with fixed constants are uniformly close whenever they lie at finite distance from one another. Every acylindrically hyperbolic group is tame by [Example 2.29] [16]. In particular, every non-elementary hyperbolic group is tame. The same authors also prove the following graphical-discreteness criterion.

Theorem 3

([Theorem 3.27] [16]). Let Λ be a finitely generated tame group. If the image of the left-multiplication homomorphism

$$\mathsf{\Lambda }⟶QI\left(\mathsf{\Lambda }\right)$$

has finite index, then Λ is graphically discrete.

Lemma 4.

Every closed subgroup of a compact-by-discrete locally compact group is compact-by-discrete.

Proof. 

Let $K⊴H$ be compact with $H/K$ discrete, and let $L\le H$ be closed. The subgroup K is open in H, so $L\cap K$ is a compact open normal subgroup of L. Thus $L/(L\cap K)$ is discrete. □

The next permanence fact concerns finite direct factors.

Lemma 5.

If F is finite and Λ is finitely generated, then $F\times \mathsf{\Lambda }$ is graphically discrete if and only if Λ is graphically discrete.

Proof. 

Suppose first that $\mathsf{\Lambda }$ is graphically discrete. The subgroup $\mathsf{\Lambda }$ has finite index in $F\times \mathsf{\Lambda }$. Restricting any geometric action of $F\times \mathsf{\Lambda }$ to $\mathsf{\Lambda }$ is still geometric, so the full automorphism group of the underlying graph is compact-by-discrete.

Conversely, if $\mathsf{\Lambda }$ has a geometric action on X witnessing failure of graphical discreteness, let $F\times \mathsf{\Lambda }$ act on the same graph through the projection to $\mathsf{\Lambda }$. Its stabilizers are $F\times {\mathsf{\Lambda }}_x$, hence finite. The action is proper and cocompact. The same graph X is therefore a witness for $F\times \mathsf{\Lambda }$. □

3. Main Results

3.1. Flexible Coxeter Systems

Let $(W,S)$ be a Coxeter system of finite rank. The standard undirected Cayley graph $\mathcal{C}(W,S)$ has vertex set W and an edge from w to $ws$ for each $w\in W$ and $s\in S$.

Definition 6

([p. 1] [20]). The defining diagram of $(W,S)$ is flexible if there are $s\in S$ and a nontrivial label-preserving automorphism $\varphi$ of the diagram such that $\varphi \left(s\right)=s$ and $\varphi \left(r\right)=r$ whenever $m_{sr}<\infty$. For $(W_{\mathsf{\Gamma }},V\mathsf{\Gamma })$, this says exactly that some nonidentity graph automorphism of $\mathsf{\Gamma }$ fixes a closed star pointwise. Thus nonflexibility in the right-angled case is precisely the star-rigid condition used in [Remark 4.12] [19].

Theorem 7.

Let $(W,S)$ be a Coxeter system with S finite. The following conditions are equivalent.

1.

$Aut\left(\mathcal{C}\right(W,S\left)\right)$ is compact-by-discrete.

2.

$Aut\left(\mathcal{C}\right(W,S\left)\right)$ is discrete.

3.

the defining diagram is nonflexible.

If the diagram is flexible, then W is not graphically discrete.

Proof. 

Assume that the diagram is nonflexible. Haglund–Paulin’s restriction isomorphism and rigid-case theorem show that the stabilizer of 1 in $Aut\left(\mathcal{C}\right(W,S\left)\right)$ is isomorphic to the finite Coxeter-diagram automorphism group $G(W,S)$ [Corollaire 5.8 and Théorème 5.12] [13]. Point stabilizers are open in the pointwise-convergence topology, so this finite stabilizer forces $Aut\left(\mathcal{C}\right(W,S\left)\right)$ to be discrete. Thus (3) implies (2). Condition (2) implies (1), since a discrete group is compact-by-discrete. It remains to show that flexibility rules out (1).

Choose s and $\varphi$ as in the definition. Since $\varphi$ is nontrivial, there is $t\in S$ such that $u=\varphi \left(t\right)\ne t$. If $m_{st}<\infty$, then $\varphi \left(t\right)=t$, a contradiction. Thus $m_{st}=\infty$. Since $\varphi \left(s\right)=s$ and $\varphi$ is injective, $u\ne s$. Label preservation then gives $m_{su}=\infty$. In particular, the flexible case forces W to be infinite.

First consider a Cayley-graph automorphism. Following White’s first-s construction [Definition 21] [20], define $\mathsf{\Psi }$ on reduced words as follows. If a reduced word contains s, write it as

$$w=w_{1}s{w}_2,$$

where $w_1$ contains no s, and set

$$\mathsf{\Psi }\left(w\right)=\varphi \left(w_1\right)s{w}_2.$$

If w contains no s, set $\mathsf{\Psi }\left(w\right)=\varphi \left(w\right)$.

First note that the displayed image word is reduced. Suppose otherwise. By Tits’ word-reduction theorem [Theorem 3.4.2] [7], choose a shortening sequence and stop at its first deletion. All preceding steps are braid moves. Pull them back across the first-s cut. A move before the cut pulls back under ${\varphi }^{-1}$, one after the cut is unchanged, and a move meeting the cut has other generator r with $m_{sr}<\infty$, hence $\varphi \left(r\right)=r$. This gives a reduced word $w^{\prime }$ braid-equivalent to w such that $\mathsf{\Psi }\left(w^{\prime }\right)$ contains an adjacent equal pair. Before the cut, ${\varphi }^{-1}$ pulls this pair back to $w^{\prime }$. After the cut it is unchanged. If it meets the cut, injectivity and $\varphi \left(s\right)=s$ force an adjacent $ss$ in $w^{\prime }$. Each case contradicts reducedness.

It remains to show that the definition is independent of the chosen reduced expression. By Matsumoto’s theorem, again [Theorem 3.4.2] [7], two reduced expressions for the same element differ by braid moves. Consider one such move. If its alternating block lies before the first s, applying $\varphi$ to the block carries the braid relation to the corresponding braid relation because $\varphi$ preserves Coxeter labels. If the block lies after the first s, the same braid move is left unchanged. Finally, if the block contains the first s, then its other letter r satisfies $m_{sr}<\infty$, so $\varphi \left(r\right)=r$. The entire block is therefore fixed by $\varphi$, even when an odd braid move changes the position of the first s. Thus $\mathsf{\Psi }$ respects every braid move and descends to a map on W. Replacing $\varphi$ by ${\varphi }^{-1}$ gives the inverse of $\mathsf{\Psi }$.

The map also preserves adjacency. If $wr$ is reduced, then either w already contains s, in which case $\mathsf{\Psi }\left(wr\right)=\mathsf{\Psi }\left(w\right)r$, or w contains no s. In the latter case, $\mathsf{\Psi }\left(wr\right)=\mathsf{\Psi }\left(w\right)s$ when $r=s$, and $\mathsf{\Psi }\left(wr\right)=\mathsf{\Psi }\left(w\right)\varphi \left(r\right)$ when $r\ne s$. If $wr$ is not reduced, apply the same argument to $w=\left(wr\right)r$. Hence $\mathsf{\Psi }$ and its inverse preserve edges, so $\mathsf{\Psi }\in Aut\left(\mathcal{C}\right(W,S\left)\right)$ and $\mathsf{\Psi }\left(1\right)=1$.

For $n\ge 0$, put

$$w_n={\left(st\right)}^{n}s,{\Phi }_n=L_{w_n}\circ \mathsf{\Psi }\circ L_{w_n}^{-1}.$$

Since $m_{st}=\infty$, the displayed word for $w_n$ is reduced and $w_n^{-1}=w_n$. Its first letter is s, so $\mathsf{\Psi }\left(w_n\right)=w_n$. Therefore each ${\Phi }_n$ fixes the identity.

For $k\ge 1$, put $y_k={\left(st\right)}^k$. If $0\le n<k$, then

$$y_k=w_{n}t{\left(st\right)}^{k-n-1},w_n^{-1}{y}_k=t{\left(st\right)}^{k-n-1}.$$

The first-s prescription gives

$$\mathsf{\Psi }\left(t{\left(st\right)}^{k-n-1}\right)=u{\left(st\right)}^{k-n-1}.$$

When $n=k-1$, the argument of $\mathsf{\Psi }$ is just t, so the formula applies as written. Consequently

$${\Phi }_n\left(y_k\right)={\left(st\right)}^n\left(su\right){\left(st\right)}^{k-n-1}.$$

The standard parabolic subgroup on $\{s,t,u\}$ splits as

$$W_{\{s,t,u\}}=\langle s\rangle *W_{\{t,u\}},$$

because no finite Coxeter relation joins s to t or u. In free-product normal form, $y_k$ has k successive $W_{\{t,u\}}$-blocks all equal to t, whereas ${\Phi }_n\left(y_k\right)$ has the $(n+1)$-st such block equal to u. Since $u\ne t$, the $k+1$ points

$$y_k,{\Phi }_0\left(y_k\right),\dots ,{\Phi }_{k-1}\left(y_k\right)$$

are pairwise distinct. Hence

$$\left|Aut{\left(\mathcal{C}(W,S)\right)}_1·y_k\right|\ge k+1(k\ge 1).$$

The full automorphism group acts cocompactly on $\mathcal{C}(W,S)$ because it contains the vertex-transitive left action of W. Theorem 2 therefore implies that $Aut\left(\mathcal{C}\right(W,S\left)\right)$ is not compact-by-discrete. Finally, the left action of W on this graph is geometric, so the same graph witnesses that W is not graphically discrete. □

3.2. Generalized Polygons and Quasi-Isometric Rigidity

By [§1.2] [19], a generalized m-gon is a connected bipartite graph of diameter m and girth $2m$, and it is thick if every vertex has degree at least three. The equivalent incidence-geometric formulation, in which ordinary m-gons are the apartments, is recalled in [§2.2] [8]. In particular, a thick generalized 3-gon is a projective plane [§1.2] [19]. By the Feit–Higman theorem, a finite thick generalized m-gon with $m\ge 3$ can occur only for $m\in \{3,4,6,8\}$ [9]. Let $\mathsf{\Gamma }$ be a finite thick generalized m-gon with $m\ge 3$. Since $\mathsf{\Gamma }$ is bipartite and hence triangle-free, its Davis complex ${\Sigma }_{\mathsf{\Gamma }}$ is a two-dimensional CAT(0) cube complex with vertex set $W_{\mathsf{\Gamma }}$ and 1-skeleton $\mathcal{C}(W_{\mathsf{\Gamma }},V\mathsf{\Gamma })$ [Proposition 7.3.4, Example 7.3.6 and Theorem 12.2.1(i)] [7].

Bounds and Xie replace every Euclidean square of ${\Sigma }_{\mathsf{\Gamma }}$ with a regular hyperbolic quadrilateral of angle $\pi /m$. The resulting proper metric space, denoted here by $X_{\mathsf{\Gamma }}$ while Bounds and Xie retain ${\Sigma }_{\mathsf{\Gamma }}$, is a Fuchsian building [Theorem 4.5] [3]. Since [Definition 2.3] [3] requires each edge to lie in at least three chambers, $X_{\mathsf{\Gamma }}$ is thick in the building-theoretic sense. The $W_{\mathsf{\Gamma }}$-action remains proper and cocompact.

Theorem 8.

Let Γ be a finite thick generalized m-gon, where $m\ge 3$, and let Λ be a finitely generated group quasi-isometric to $W_{\mathsf{\Gamma }}$. Then

$$\mathsf{\Lambda }\mathit{is}\mathit{graphically}\mathit{discrete}⟺\mathsf{\Gamma }\mathit{is}\mathit{nonflexible}.$$

Lemma 9.

Every metric isometry of $X_{\mathsf{\Gamma }}$ preserves the cubical vertices, the open cubical edges and the open squares. Restriction to the vertex set therefore defines a continuous injective homomorphism

$$Isom\left(X_{\mathsf{\Gamma }}\right)⟶Aut\left(\mathcal{C}(W_{\mathsf{\Gamma }},V\mathsf{\Gamma })\right).$$

Proof. 

The three metric strata are distinguished by their spaces of directions. At an interior point of a square, the space of directions is a circle. If a cubical edge has type $s\in V\mathsf{\Gamma }$, then it is contained in one square for each $t\in {lk}_{\mathsf{\Gamma }}\left(s\right)$. Hence an interior point of that edge has space of directions the suspension of a ${deg}_{\mathsf{\Gamma }}\left(s\right)$-point set. Thickness gives ${deg}_{\mathsf{\Gamma }}\left(s\right)\ge 3$, so this is a graph with exactly two branch points. At a cubical vertex, the space of directions is the metric realization of $\mathsf{\Gamma }$, with every edge of length $\pi /m$. The graph $\mathsf{\Gamma }$ has more than two vertices and every one of them has degree at least three, so this link has more than two branch points. Thus the three strata are metric invariants.

It follows that every isometry preserves the cubical vertices and open edges. The connected components of the complement of those strata are the open squares. A cubical square is determined by its four vertices, so an isometry fixing every cubical vertex preserves every square setwise. Its restriction to a regular hyperbolic square fixes all four vertices and is therefore the identity on that square. Restriction to the vertex set is thus injective. The cubical vertex set is locally finite and therefore discrete. For every finite set of vertices A, there is $\epsilon >0$ such that an isometry moving each point of A by less than $\epsilon$ fixes A pointwise. Hence the preimage of every basic pointwise stabilizer in the target is open in the compact-open topology, which proves continuity. □

Proof of Theorem 8.

By Bounds–Xie quasi-isometric rigidity, $\mathsf{\Lambda }$ acts geometrically on $X_{\mathsf{\Gamma }}$ [Corollary 1.2] [3]. Lemma 9 shows that this action preserves the cubical vertex and edge strata and hence restricts to an action on the standard Cayley graph $\mathcal{C}(W_{\mathsf{\Gamma }},V\mathsf{\Gamma })$. This restricted action is proper. Finite vertex sets are compact subsets of $X_{\mathsf{\Gamma }}$, so their transporter sets are finite for the original proper action of the discrete group $\mathsf{\Lambda }$. It is also cocompact. Indeed, if $K\subseteq X_{\mathsf{\Gamma }}$ is compact and $\mathsf{\Lambda }K=X_{\mathsf{\Gamma }}$, then K meets the locally finite cubical vertex set in only finitely many points. Since $\mathsf{\Lambda }$ preserves that vertex set, those points meet every $\mathsf{\Lambda }$-orbit of cubical vertices.

If $\mathsf{\Gamma }$ is flexible, Theorem 7 shows that $Aut\left(\mathcal{C}(W_{\mathsf{\Gamma }},V\mathsf{\Gamma })\right)$ is not compact-by-discrete. The preceding geometric action therefore witnesses that $\mathsf{\Lambda }$ is not graphically discrete.

Assume that $\mathsf{\Gamma }$ is nonflexible. By Theorem 7, $Aut\left(\mathcal{C}(W_{\mathsf{\Gamma }},V\mathsf{\Gamma })\right)$ is discrete. Lemma 9 then implies that $G=Isom\left(X_{\mathsf{\Gamma }}\right)$ is discrete. Let $J\le G$ be the image of the action homomorphism $\mathsf{\Lambda }\to G$. Properness gives finite point stabilizers, hence a finite kernel. Moreover, J has finite index in G. Indeed, choose a compact set $K\subseteq X_{\mathsf{\Gamma }}$ with $JK=X_{\mathsf{\Gamma }}$, and fix $x\in X_{\mathsf{\Gamma }}$. Since $X_{\mathsf{\Gamma }}$ is proper, the full isometry group acts properly, so

$$F=\{g\in G:gx\in K\}$$

is compact and hence finite because G is discrete. For each $g\in G$, choose $j\in J$ with $j^{-1}gx\in K$. Then $j^{-1}g\in F$, so $G=JF$.

Every self-quasi-isometry of $X_{\mathsf{\Gamma }}$ is at finite distance from an isometry [Theorem 1.1] [3]. Hence the natural map $G\to QI\left(X_{\mathsf{\Gamma }}\right)$ is surjective. Since J has finite index in G, its image has finite index in $QI\left(X_{\mathsf{\Gamma }}\right)$. An orbit map $o:\mathsf{\Lambda }\to X_{\mathsf{\Gamma }}$ is a quasi-isometry [p. 58] [16], and $o\circ L_{\lambda }=\lambda \circ o$ for left multiplication $L_{\lambda }$. Under the isomorphism $QI\left(\mathsf{\Lambda }\right)\cong QI\left(X_{\mathsf{\Gamma }}\right)$ induced by o, the image of the left-multiplication homomorphism $\mathsf{\Lambda }\to QI\left(\mathsf{\Lambda }\right)$ is the image of J. It therefore has finite index.

Finally, $\mathsf{\Gamma }$ has no induced four-cycle, so $W_{\mathsf{\Gamma }}$ is hyperbolic [Corollary 12.6.3] [7]. Each vertex has three pairwise nonadjacent neighbours, which generate a special subgroup $C_2*C_2*C_2$. Hence $W_{\mathsf{\Gamma }}$ is non-elementary. Consequently every finitely generated group quasi-isometric to $W_{\mathsf{\Gamma }}$, including $\mathsf{\Lambda }$, is non-elementary hyperbolic and hence tame [Example 2.29] [16]. Theorem 3 completes the proof. □

Remark 10.

Corollary 4.2 of [16] states that uniform lattices in thick locally finite hyperbolic buildings are not graphically discrete. There is therefore a conditional tension with Theorem 8. For a finite thick generalized m-gon $\mathsf{\Gamma }$, the standard $W_{\mathsf{\Gamma }}$-action on $X_{\mathsf{\Gamma }}$ is faithful and geometric. Since $X_{\mathsf{\Gamma }}$ is proper, [Lemma 2.24] [16] realizes $W_{\mathsf{\Gamma }}$ as a uniform lattice in $Isom\left(X_{\mathsf{\Gamma }}\right)$. Hence, if Corollary 4.2 holds as stated, applying Theorem 8 with $\mathsf{\Lambda }=W_{\mathsf{\Gamma }}$ forces every finite thick generalized m-gon with $m\ge 3$ to be flexible. Equivalently, the existence of a nonflexible such $\mathsf{\Gamma }$ would give a counterexample to Corollary 4.2. In particular, by Proposition 18, Corollary 4.2 would imply that every finite projective plane of order at least two admits a nontrivial perspectivity. Thus Corollary 4.2 and an affirmative answer to Question 27 cannot both hold.

The displayed proof of Corollary 4.2 uses the assertion that any two apartments containing a fixed chamber can be interchanged by a global building automorphism fixing that chamber pointwise. The standard building axiom recalled in [§2] [21] supplies an isomorphism between two apartments fixing their intersection. By itself, it does not supply an extension of that apartment isomorphism to a global building automorphism. Thus the displayed proof requires an additional apartment-extension assumption. This observation concerns that proof only. No claim about the truth value of Corollary 4.2 is made here.

The proof of Theorem 8 is independent of Corollary 4.2 and of any apartment-extension property. It uses [Corollary 1.2] [3], Lemma 9, [Theorem 1.1] [3], and [Theorem 3.27] [16].

Proposition 11.

The incidence graph is flexible for every finite thick Moufang generalized polygon and for every finite thick generalized quadrangle that is an elation generalized quadrangle or the dual of one.

Proof. 

For the Moufang case, let $\alpha =(x_0,\dots ,x_m)$ be a root. The root group $U_{\alpha }$ consists of the elations fixing every element incident with one of the interior vertices $x_1,\dots ,x_{m-1}$ [§3] [8]. For a thick Moufang polygon, $U_{\alpha }$ is nontrivial and acts regularly on the apartments through $\alpha$ [§3] [8]. Hence any nonidentity element of $U_{\alpha }$ fixes the closed star of each interior vertex $x_i$ pointwise.

For the elation case, let p be the base point and let E be an elation group about p. By definition, E fixes every line through p and acts regularly on the points not collinear with p [Chapter 8, pp. 138–139] [18]. If the quadrangle has order $(s,t)$, the set of points not collinear with p has cardinality $s^{2}t$. Thickness gives $s,t\ge 2$, so this cardinality is at least eight. Hence E is nontrivial. A nonidentity element of E fixes the closed star of the point vertex p pointwise. Interchanging points and lines gives the dual statement. □

Consequently, any example realizing the graphically discrete case of Theorem 8 must be non-Moufang. In the quadrangle case it can be neither an elation generalized quadrangle nor the dual of one. These restrictions do not amount to a classification of all finite generalized polygons.

3.3. Coxeter Doublings and Finite-Index Witnesses

Throughout this subsection, $\mathsf{\Gamma }$ is a finite simplicial graph. For a vertex $v\in V\mathsf{\Gamma }$, write

$${st}_{\mathsf{\Gamma }}\left(v\right)=\left\{v\right\}\cup {lk}_{\mathsf{\Gamma }}\left(v\right)$$

for its closed star.

Fix $v\in V\mathsf{\Gamma }$. The double$D_v\mathsf{\Gamma }$ consists of two copies of $\mathsf{\Gamma }\setminus \left\{v\right\}$ glued along ${lk}_{\mathsf{\Gamma }}\left(v\right)$ [§2.2, immediately before Lemma 2.7] [6]. Equivalently, vertices in the link of v have one copy, while every vertex outside ${st}_{\mathsf{\Gamma }}\left(v\right)$ has two copies. Define

$${\epsilon }_v:W_{\mathsf{\Gamma }}⟶C_2$$

by sending v to the nontrivial element and every other standard generator to the identity, and put $H_v:=ker\left({\epsilon }_v\right)$. The Doubling Lemma [Lemma 2.7] [6] states that $W_{D_v\mathsf{\Gamma }}$ occurs as an index-two subgroup of $W_{\mathsf{\Gamma }}$. For the explicit realization used below, apply Reidemeister–Schreier with transversal $\{1,v\}$. For each $r\ne v$ the resulting generators are r and $r^{*}=vrv$. They coincide when $r\in {lk}_{\mathsf{\Gamma }}\left(v\right)$. If $r\notin {st}_{\mathsf{\Gamma }}\left(v\right)$, the special subgroup $W_{\{v,r\}}\cong C_2*C_2$ shows that $r^{*}\ne r$. Rewriting the involution and commutation relators gives

$$r^2={\left(r^{*}\right)}^2=1,[r,s]=[r^{*},s^{*}]=1\mathrm{for}\{r,s\}\in E\mathsf{\Gamma },r,s\ne v.$$

When a vertex lies in ${lk}_{\mathsf{\Gamma }}\left(v\right)$, the equality $r^{*}=r$ makes the link common to the two copies. No cross-copy commutation relation occurs between two vertices outside the closed star. Thus this is exactly the Coxeter presentation of $W_{D_v\mathsf{\Gamma }}$. Hence $H_v\cong W_{D_v\mathsf{\Gamma }}$, with a link vertex represented by r and the two copies of each $r\notin {st}_{\mathsf{\Gamma }}\left(v\right)$ represented by r and $vrv$.

Theorem 12.

Let Γ be a finite simplicial graph and let $v\in V\mathsf{\Gamma }$. If $D_v\mathsf{\Gamma }$ is flexible, then $W_{\mathsf{\Gamma }}$ is not graphically discrete.

Proof. 

Let

$$X=\mathcal{C}\left(W_{D_v\mathsf{\Gamma }},V\left(D_v\mathsf{\Gamma }\right)\right)$$

be the standard Cayley graph of $H_v$. Conjugation by v fixes each standard generator from ${lk}_{\mathsf{\Gamma }}\left(v\right)$ and interchanges the two generators r and $vrv$ associated to each vertex $r\notin {st}_{\mathsf{\Gamma }}\left(v\right)$. It therefore induces the deck involution of $D_v\mathsf{\Gamma }$ and an automorphism $\alpha$ of X.

The splitting $W_{\mathsf{\Gamma }}=H_v⋊\langle v\rangle$ now gives an action on X by

$$\left(h{v}^{ϵ}\right)·x=h{\alpha }^{ϵ}\left(x\right)(h,x\in H_v,ϵ\in \{0,1\}).$$

The subgroup $H_v$ acts simply transitively on the vertices, and the stabilizer of the identity in $W_{\mathsf{\Gamma }}$ is $\langle v\rangle$. Thus the action of $W_{\mathsf{\Gamma }}$ on X is proper and cocompact.

If $D_v\mathsf{\Gamma }$ is flexible, Theorem 7 implies that $Aut\left(X\right)$ is not compact-by-discrete. Hence this geometric action on X witnesses that $W_{\mathsf{\Gamma }}$ is not graphically discrete. □

For a component C of $\mathsf{\Gamma }\setminus {st}_{\mathsf{\Gamma }}\left(v\right)$, let ${\tau }_C$ interchange the two copies of C in $D_v\mathsf{\Gamma }$ and fix every other vertex. Since distinct components have no edges between them, ${\tau }_C$ is a graph automorphism.

Corollary 13.

If $\mathsf{\Gamma }\setminus {st}_{\mathsf{\Gamma }}\left(v\right)$ has at least two connected components for some $v\in V\mathsf{\Gamma }$, then $W_{\mathsf{\Gamma }}$ is not graphically discrete.

Proof. 

Choose distinct components C and $C^{\prime }$ and a vertex r in one copy of $C^{\prime }$. Every vertex in the closed star of r in $D_v\mathsf{\Gamma }$ belongs either to that same copy of $C^{\prime }$ or to the common link of v. The nontrivial automorphism ${\tau }_C$ fixes all of these vertices. Thus $D_v\mathsf{\Gamma }$ is flexible, and Theorem 12 applies. □

The component swap also has a group-theoretic interpretation. The standard partial conjugation supported on C is the automorphism described in [p. 1747] [11], namely

$${\chi }_{v,C}\left(r\right)=\left\{\begin{array}{cc}vrv,\hfill & r\in C,\hfill \\ r,\hfill & r\notin C\hfill \end{array}\right.(r\in V\mathsf{\Gamma }).$$

Its restriction to $H_v$ induces ${\tau }_C$ on the doubled Coxeter generating set. The product of the partial conjugations over all components is conjugation by v. When there are at least two components, each individual factor has proper support. Corollary 13 therefore shows that a proper partial conjugation obstructs graphical discreteness.

A similar component swap can be used in some cases even when the complement of the star is connected.

Corollary 14.

Suppose that C is a connected component of $\mathsf{\Gamma }\setminus {st}_{\mathsf{\Gamma }}\left(v\right)$ and that some $s\in {lk}_{\mathsf{\Gamma }}\left(v\right)$ has no neighbour in C. Then $W_{\mathsf{\Gamma }}$ is not graphically discrete.

Proof. 

In $D_v\mathsf{\Gamma }$, interchange the two copies of C and fix every other vertex. There are no edges from C to another component of $\mathsf{\Gamma }\setminus {st}_{\mathsf{\Gamma }}\left(v\right)$, and its two copies have identical adjacency to the common link of v. The interchange is therefore a nontrivial graph automorphism. It fixes the closed star of s pointwise because s has no neighbour in C. Thus $D_v\mathsf{\Gamma }$ is flexible, and Theorem 12 applies. □

Corollary 15.

If distinct vertices $s,v\in V\mathsf{\Gamma }$ satisfy

$${st}_{\mathsf{\Gamma }}\left(s\right)\subseteq {st}_{\mathsf{\Gamma }}\left(v\right)\mathrm{and}{st}_{\mathsf{\Gamma }}\left(v\right)\ne V\mathsf{\Gamma },$$

then $W_{\mathsf{\Gamma }}$ is not graphically discrete.

Proof. 

The star inclusion implies that $s\in {lk}_{\mathsf{\Gamma }}\left(v\right)$ and that s has no neighbour outside ${st}_{\mathsf{\Gamma }}\left(v\right)$. Since ${st}_{\mathsf{\Gamma }}\left(v\right)\ne V\mathsf{\Gamma }$, the complement has a vertex. Apply Corollary 14 to any component of $\mathsf{\Gamma }\setminus {st}_{\mathsf{\Gamma }}\left(v\right)$. □

Corollary 13 detects examples missed by the link-gap criterion in Corollary 14, even when the defining graph is asymmetric with no dominated vertices and the associated right-angled Coxeter group is one-ended and hyperbolic. For $m\ge 2$, let ${\mathsf{\Theta }}_m$ be the union of the paths

$$P_0=s-v-t,P=s-a_1-a_2-a_3-a_4-a_5-t,Q_m=s-b_1-\dots -b_m-t,$$

and the additional path

$$R=a_1-c_1-c_2-c_3-a_4,$$

where all displayed internal vertices are distinct. The graph ${\mathsf{\Theta }}_m$ is connected by construction.

Corollary 16.

For $m\ge 2$, the graph ${\mathsf{\Theta }}_m$ is asymmetric and has no distinct vertices $x,y$ satisfying ${st}_{{\mathsf{\Theta }}_m}\left(x\right)\subseteq {st}_{{\mathsf{\Theta }}_m}\left(y\right)$. The groups $W_{{\mathsf{\Theta }}_m}$ are pairwise nonisomorphic, one-ended, hyperbolic and not graphically discrete. Moreover, there is no triple $(x,C,y)$ in which C is a component of ${\mathsf{\Theta }}_m\setminus {st}_{{\mathsf{\Theta }}_m}\left(x\right)$ and $y\in {lk}_{{\mathsf{\Theta }}_m}\left(x\right)$ has no neighbour in C.

Proof. 

The vertices of degree three are precisely $s,t,a_1,a_4$. Every other vertex has degree two. Suppressing the degree-two vertices gives a four-vertex multigraph. The two parallel s–t paths have lengths 2 and $m+1$, whereas the two parallel $a_1$–$a_4$ paths have lengths 3 and 4. These two unordered pairs of lengths are distinct, so every automorphism preserves the unordered pairs $\{s,t\}$ and $\{a_1,a_4\}$. The two remaining paths joining these pairs have lengths one and two. The unique length-one path singles out s and $a_1$. Hence every automorphism fixes all four branch vertices. The parallel paths have distinct lengths, so each is preserved and all of its internal vertices are fixed. Thus ${\mathsf{\Theta }}_m$ is asymmetric.

The cycles obtained from the two parallel $a_1$–$a_4$ paths and the two parallel s–t paths have lengths 7 and $m+3$, respectively. Every remaining cycle uses the two single paths and one path from each parallel pair, and hence has length 8, 9, $m+7$ or $m+8$. Thus ${\mathsf{\Theta }}_m$ has girth $min\{7,m+3\}\ge 5$. In particular it is triangle-free and has no induced four-cycle, so $W_{{\mathsf{\Theta }}_m}$ is hyperbolic [Corollary 12.6.3] [7].

Every vertex has degree at least two. If $st\left(x\right)\subseteq st\left(y\right)$ for distinct vertices, then x and y are adjacent. A second neighbour z of x would also be adjacent to y, producing a triangle. This proves that there are no dominated vertices.

There is no separating clique. Since the graph is triangle-free, every clique is a vertex or an edge. First consider deleting a vertex. If the deleted vertex is internal to one of the six maximal branch-to-branch paths, that path is replaced by at most two tails attached to its branch endpoints. The suppressed multigraph remains connected after removing any one of its six paths. If instead a branch vertex is deleted, the other branch vertices that survive remain connected in the suppressed multigraph, and every remnant of a path incident with the deleted vertex is a tail attached to its opposite endpoint. Thus no vertex separates ${\mathsf{\Theta }}_m$.

Next consider deleting the endpoints of an edge lying on a maximal path J. If neither endpoint is a branch vertex, the surviving pieces of J are tails attached to the endpoints of J, while the suppressed multigraph with J removed is connected. If exactly one endpoint is a branch vertex, delete that branch vertex in the suppressed multigraph. The remaining branch core is connected, and all surviving path pieces attach to it. The only edge whose two endpoints are branch vertices is $s-a_1$. After deleting s and $a_1$, the vertices t and $a_4$ remain joined by the path through $a_5$, and every surviving segment attaches to this connected core. Hence deleting a clique never disconnects ${\mathsf{\Theta }}_m$. In a right-angled Coxeter group, every clique generates a finite special subgroup. Therefore [Corollary 16] [17] implies that $W_{{\mathsf{\Theta }}_m}$ has at most one end. Since s and t are nonadjacent, the special subgroup $W_{\{s,t\}}\cong C_2*C_2$ is infinite. Thus $W_{{\mathsf{\Theta }}_m}$ is infinite and hence one-ended.

For every $x\ne v$, the graph ${\mathsf{\Theta }}_m\setminus {st}_{{\mathsf{\Theta }}_m}\left(x\right)$ is connected. Suppose first that x is internal to a maximal path. If neither neighbour of x is a branch vertex, removing the closed star of x leaves at most two tails attached to the connected branch core obtained by deleting that maximal path. If exactly one neighbour is a branch vertex, remove that branch vertex from the suppressed multigraph. The remaining branch core is connected, and all surviving path pieces attach to it. The only internal vertex other than v adjacent to two branch vertices is $a_5$. Deleting $st\left(a_5\right)=\{a_4,a_5,t\}$ leaves s joined to $a_1$, with all surviving path segments attached to that pair. If x is a branch vertex, the surviving core can be described directly. For $x=s$, the vertices t and $a_4$ remain joined through $a_5$. For $x=t$, the vertices s and $a_1$ remain adjacent. For $x=a_1$, the vertices t and $a_4$ remain joined through $a_5$, while for $x=a_4$, the vertices $s,t,a_1$ remain connected. Every remaining path segment attaches to the indicated core. Since ${\mathsf{\Theta }}_m$ is triangle-free and has minimum degree two, every $y\in lk\left(x\right)$ has a second neighbour outside $st\left(x\right)$, and hence a neighbour in that unique component. For $x=v$, the complement of the closed star has exactly two components,

$$A=\{a_1,\dots ,a_5,c_1,c_2,c_3\},B=\{b_1,\dots ,b_m\},$$

and each of the two link vertices meets both components. The vertex s is adjacent to $a_1,b_1$, and t is adjacent to $a_5,b_m$. This proves the final assertion and shows that Corollary 14 does not detect the family.

On the other hand, the displayed sets A and B are distinct components of ${\mathsf{\Theta }}_m\setminus st\left(v\right)$, so Corollary 13 proves that $W_{{\mathsf{\Theta }}_m}$ is not graphically discrete. Finally, $W_{{\mathsf{\Theta }}_m}^{\mathrm{ab}}\cong {\left(C_2\right)}^{m+11}$, and hence the groups are pairwise nonisomorphic. □

3.4. Projective Planes

For projective planes, nonflexibility imposes additional structure. The first result describes the automorphism groups of one-vertex doubles.

Proposition 17.

Let Γ be the incidence graph of a finite projective plane of order $q\ge 2$, let $v\in V\mathsf{\Gamma }$, and let ${\delta }_v$ be the deck involution of $D_v\mathsf{\Gamma }$. If Γ is nonflexible, then

$$Aut\left(D_v\mathsf{\Gamma }\right)=Aut{\left(\mathsf{\Gamma }\right)}_v\times \langle {\delta }_v\rangle ,$$

where $Aut{\left(\mathsf{\Gamma }\right)}_v$ denotes the stabilizer of v and acts diagonally on the two copies. In particular, $D_v\mathsf{\Gamma }$ is nonflexible.

Proof. 

Fix v and put $L={lk}_{\mathsf{\Gamma }}\left(v\right)$. In $D_v\mathsf{\Gamma }$, every vertex of L has degree $2q$, while every other vertex has degree $q+1$. Since $q\ge 2$, the set L is invariant under $Aut\left(D_v\mathsf{\Gamma }\right)$.

The graph $\mathsf{\Gamma }\setminus {st}_{\mathsf{\Gamma }}\left(v\right)$ is connected. By point-line duality, assume that v is a point. For distinct points $x,y\ne v$, if $x,y,v$ are not collinear, the path $x-\left(xy\right)-y$ avoids the star of v. If $x,y,v$ are collinear, choose a point z outside their common line. Then

$$x-\left(xz\right)-z-\left(zy\right)-y$$

is such a path. Every line not through v is incident with a remaining point. Thus deleting L from $D_v\mathsf{\Gamma }$ leaves exactly two connected components, the two copies of $\mathsf{\Gamma }\setminus {st}_{\mathsf{\Gamma }}\left(v\right)$.

After composing with ${\delta }_v$ if necessary, every automorphism may be assumed to preserve the two components. Since $D_v\mathsf{\Gamma }$ is connected and bipartite, an automorphism either preserves or interchanges its two bipartition classes. The nonempty invariant set L lies in one class, so the classes are preserved. After restoring v, the induced automorphisms give elements $g_1,g_2\in Aut{\left(\mathsf{\Gamma }\right)}_v$ with the same restriction to L. Hence $g_2^{-1}{g}_1$ fixes ${st}_{\mathsf{\Gamma }}\left(v\right)$ pointwise, and nonflexibility gives $g_1=g_2$. Consequently

$$Aut\left(D_v\mathsf{\Gamma }\right)=Aut{\left(\mathsf{\Gamma }\right)}_v\times \langle {\delta }_v\rangle ,$$

where the first factor acts diagonally. The deck involution commutes with this diagonal action and does not belong to it, so the product is direct.

Suppose an automorphism fixes the closed star of a vertex r in $D_v\mathsf{\Gamma }$ pointwise. Its decomposition cannot involve ${\delta }_v$. If $r\notin L$, the deck involution interchanges the components. If $r\in L$, it interchanges the two sets of neighbours of r outside L. The automorphism is therefore diagonal, induced by some $g\in Aut{\left(\mathsf{\Gamma }\right)}_v$. If $r\notin L$, then g fixes the corresponding closed star in $\mathsf{\Gamma }$. If $r\in L$, it fixes all neighbours of r other than v and also fixes v, so again it fixes a closed star in $\mathsf{\Gamma }$. Nonflexibility forces $g=1$. □

Proposition 18.

Let Π be a finite projective plane with incidence graph Γ. Then Γ is flexible if and only if Π admits a nontrivial perspectivity.

Proof. 

A nontrivial perspectivity with centre p fixes p and every line through p, hence fixes the closed star of the point vertex p in $\mathsf{\Gamma }$. Thus $\mathsf{\Gamma }$ is flexible.

Conversely, let $1\ne g\in Aut\left(\mathsf{\Gamma }\right)$ fix ${st}_{\mathsf{\Gamma }}\left(v\right)$ pointwise. Since $\mathsf{\Gamma }$ is connected and bipartite and g fixes v, the automorphism g preserves the point and line classes and hence induces a collineation of $\mathsf{\Pi }$. If v is a point, then g fixes every line through v, so v is a centre of g. If v is a line, then v is an axis of g. The standard centre-axis theorem for projective-plane collineations states that a collineation has a centre if and only if it has an axis [Theorem 4.9, p. 94] [14]. Hence g is a nontrivial central collineation, or equivalently an axial collineation, and therefore a perspectivity. □

Corollary 19.

If a finite projective plane of order $q\ge 2$ is perspectivity-free and has incidence graph Γ, then $W_{\mathsf{\Gamma }}$ is graphically discrete and every one-vertex double $D_v\mathsf{\Gamma }$ is nonflexible.

Proof. 

By Proposition 18, $\mathsf{\Gamma }$ is nonflexible. Apply Theorem 8 and Proposition 17. □

Let $\mathsf{\Pi }$ be a finite projective plane and let $Col\left(\mathsf{\Pi }\right)$ denote its collineation group.

Proposition 20.

Let Π have order q and incidence graph Γ. If Γ is nonflexible (equivalently, Π is perspectivity-free) and q is not a perfect square, then $Col\left(\mathsf{\Pi }\right)$ has odd order and is solvable.

Proof. 

By Proposition 18, $\mathsf{\Pi }$ has no nontrivial perspectivity. By Baer’s involution theorem, an involution of a finite projective plane is either a perspectivity or fixes a subplane of order $\sqrt{q}$ pointwise [p. 878] [5]. The latter case can occur only when q is a perfect square. Thus, under the stated hypotheses, $Col\left(\mathsf{\Pi }\right)$ contains no involution. Cauchy’s theorem then implies that $|Col(\mathsf{\Pi }\left)\right|$ is odd. The Feit–Thompson theorem shows that $Col\left(\mathsf{\Pi }\right)$ is solvable [Chapter I, §1, p. 775] [10]. □

Equivalently, a projective plane of nonsquare order with even-order collineation group has a flexible incidence graph, so the corresponding right-angled Coxeter group is not graphically discrete. Hence a projective plane with a nonflexible incidence graph has either square order or nonsquare order with an odd solvable collineation group. Proposition 20 therefore narrows the finite-geometric search but does not resolve the existence question.

3.5. Unbounded Orbits from Thick Panels

Let X be a locally finite semi-regular right-angled building of type $(W,S)$, where S is finite. Here semi-regular means that every r-panel has the same finite cardinality $q_r\ge 2$, depending only on $r\in S$ [§1] [4]. Write $Aut{\left(X\right)}^{+}$ for the type-preserving automorphism group. A building automorphism is determined by its action on chambers and may therefore be viewed as the induced chamber-graph automorphism. Because S is finite, preservation of each chamber-adjacency type is a closed condition in the pointwise-convergence topology. Hence $Aut{\left(X\right)}^{+}$ is a closed subgroup of the automorphism group of the chamber graph. It is strongly transitive, and hence chamber-transitive, by [Proposition 6.1] [4].

For a panel $\sigma$, write $Ch\left(\sigma \right)$ for its set of chambers. For a chamber x, write ${proj}_{\sigma }\left(x\right)$ for the gate projection of x to $\sigma$. Caprace’s extension theorem [Proposition 4.2] [4] states that every permutation $\alpha$ of the chambers of a panel $\sigma$ extends to an element of $Aut{\left(X\right)}^{+}$ that stabilizes $\sigma$, induces $\alpha$ on it, and fixes every chamber whose projection to $\sigma$ is fixed by $\alpha$. Both propositions are stated for semi-regular right-angled buildings and do not require $q_r>2$ for every $r\in S$.

Under the rank-two hypothesis below, Kubena and Thomas proved that both the type-preserving and full automorphism groups are nondiscrete [Theorem 1(1)] [15]. The next proposition strengthens the type-preserving conclusion to failure of compact-by-discreteness by exhibiting unbounded chamber-stabilizer orbits.

Proposition 21.

Suppose that $q_s>2$ and $m_{st}=\infty$ for some $s,t\in S$. Then $Aut{\left(X\right)}^{+}$ is not compact-by-discrete.

Proof. 

Fix a chamber c and $k\ge 1$. The incidence graph of the s- and t-panels in the rank-two residue through c is a tree [§1] [4]. Choose a chamber y so that a minimal gallery from c to y crosses, in order, distinct s-panels

$${\sigma }_1,\dots ,{\sigma }_k.$$

Put

$$a_i={proj}_{{\sigma }_i}\left(c\right),b_i={proj}_{{\sigma }_i}\left(y\right).$$

The two projections $a_i$ and $b_i$ are distinct. Since $q_s>2$, choose $e_i\in Ch\left({\sigma }_i\right)\setminus \{a_i,b_i\}$.

Let ${\alpha }_i$ be the transposition of $b_i$ and $e_i$ fixing every other chamber of ${\sigma }_i$. Extend it by Caprace’s theorem to $g_i\in Aut{\left(X\right)}^{+}$. Since ${\alpha }_i\left(a_i\right)=a_i$, the automorphism $g_i$ fixes c.

If $i<j$, the unique path in the rank-two incidence tree shows that every chamber of ${\sigma }_i$ projects to $a_j$ on ${\sigma }_j$. Hence $g_j$ fixes ${\sigma }_i$ pointwise. Equivariance of gate projections gives

$${proj}_{{\sigma }_i}\left(g_{i}y\right)=e_i,{proj}_{{\sigma }_i}\left(g_{j}y\right)=b_i.$$

Thus $g_{i}y\ne g_{j}y$ whenever $i<j$. Moreover, ${proj}_{{\sigma }_i}\left(g_{i}y\right)=e_i\ne b_i={proj}_{{\sigma }_i}\left(y\right)$, so $g_{i}y\ne y$ for every i. Consequently

$${|Aut\left(X\right)}_c^{+}·y|\ge k+1.$$

The group $Aut{\left(X\right)}^{+}$ is closed and chamber-transitive, hence acts cocompactly on the connected locally finite chamber graph. Since k is arbitrary, Theorem 2 proves the proposition. □

3.6. The Standard Chamber Graph of a Graph Product

Let $\mathsf{\Gamma }$ be a finite simplicial graph with vertex set S, and let ${\left(G_s\right)}_{s\in S}$ be nontrivial finite groups. Their graph product [Chapter 3] [12] is

$$G_{\mathsf{\Gamma }}=\left({*}_{s\in S}{G}_s\right)/\langle \langle [G_s,G_t]:\{s,t\}\in E\mathsf{\Gamma }\rangle \rangle .$$

By Shepherd’s construction [Definitions 2.6 and 3.1, Lemmas 2.5(1) and 2.10] [19], the standard chamber graph ${\mathcal{C}}_{\mathsf{\Gamma }}$ has vertex set $G_{\mathsf{\Gamma }}$. Two distinct vertices $g,h$ are adjacent exactly when $g^{-1}h\in G_s\setminus \left\{1\right\}$ for some $s\in S$. There is an associated right-angled building ${\mathcal{B}}_{\mathsf{\Gamma }}$. Its chamber graph is ${\mathcal{C}}_{\mathsf{\Gamma }}$, its s-panels have cardinality $q_s:=\left|G_s\right|$, and $G_{\mathsf{\Gamma }}$ acts simply transitively on its chambers.

The discrete direction rests on the following elementary product observation.

Lemma 22.

Let A and B be connected graphs. Suppose every edge of A lies in a triangle and B is triangle-free. With the pointwise-convergence topologies, the map

$$(\alpha ,\beta )⟼\left((a,b)⟼\left(\alpha \right(a),\beta (b\left)\right)\right)$$

is an isomorphism of topological groups from $Aut\left(A\right)\times Aut\left(B\right)$ onto $Aut(A\square B)$. In particular, the images of both factors are closed in the product automorphism group. If A is finite and $Aut\left(B\right)$ is discrete, then $Aut(A\square B)$ is discrete.

Proof. 

If $A=K_1$ or $B=K_1$, the assertion is immediate, so assume both factors have an edge. Every triangle in a Cartesian product lies in a layer of one factor. Hence an edge of $A\square B$ lies in a triangle if and only if it is an A-edge. Every automorphism therefore preserves the A- and B-edges separately. Since A and B are connected, the corresponding layers are the connected components of the two single-edge-type subgraphs, so both families of layers are permuted.

For $f\in Aut(A\square B)$, write

$$f(A\times \{b\left\}\right)=A\times \left\{\beta \right(b\left)\right\},f\left(\right\{a\}\times B)=\left\{\alpha \right(a\left)\right\}\times B.$$

Intersecting the two image layers gives

$$f(a,b)=\left(\alpha \right(a),\beta (b\left)\right).$$

Preservation of the two edge types shows that $\alpha \in Aut\left(A\right)$ and $\beta \in Aut\left(B\right)$. Conversely, every pair of factor automorphisms acts on the product, so the displayed map is an abstract group isomorphism.

It remains to check the topology. The forward map is continuous by coordinate evaluation. Fix $a_0\in V\left(A\right)$ and $b_0\in V\left(B\right)$. For an automorphism f of the product, the value $\alpha \left(a\right)$ is the first coordinate of $f(a,b_0)$, and $\beta \left(b\right)$ is the second coordinate of $f(a_0,b)$. Thus the inverse is continuous for the pointwise-convergence topologies. The map is therefore a topological group isomorphism, and its two factors are closed. If A is finite, then $Aut\left(A\right)$ is finite, giving the last assertion. □

In view of the Kubena–Thomas discreteness theorem [Theorem 1] [15], it remains to determine when compact-by-discreteness can occur for the standard chamber graph.

Theorem 23.

Set

$$T=\{s\in S:|G_s|>2\},{\mathsf{\Gamma }}_0=\mathsf{\Gamma }[S\setminus T].$$

The following conditions are equivalent.

1.

$Aut\left({\mathcal{C}}_{\mathsf{\Gamma }}\right)$ is compact-by-discrete.

2.

$Aut\left({\mathcal{C}}_{\mathsf{\Gamma }}\right)$ is discrete.

3.

every vertex of T is universal in Γ and ${\mathsf{\Gamma }}_0$ is nonflexible.

Proof. 

Suppose first that $s\in T$ is not universal, and choose $t\ne s$ not adjacent to s. Then $q_s>2$ and $m_{st}=\infty$ in $W_{\mathsf{\Gamma }}$. Proposition 21 shows that the closed subgroup $Aut{\left({\mathcal{B}}_{\mathsf{\Gamma }}\right)}^{+}$ of $Aut\left({\mathcal{C}}_{\mathsf{\Gamma }}\right)$ is not compact-by-discrete. Lemma 4 therefore implies that $Aut\left({\mathcal{C}}_{\mathsf{\Gamma }}\right)$ is not compact-by-discrete.

Assume now that every vertex of T is universal. The graph product splits as a direct product, and the standard generating set is the disjoint union of the generating sets of its two factors. Hence its standard chamber graph is the Cartesian product

$$G_{\mathsf{\Gamma }}=\left(\prod _{s\in T}{G}_s\right)\times W_{{\mathsf{\Gamma }}_0},{\mathcal{C}}_{\mathsf{\Gamma }}=A\square B,$$

where

$$A=\underset{s\in T}{\square }{K}_{q_s},B=\mathcal{C}(W_{{\mathsf{\Gamma }}_0},S\setminus T).$$

If $T=⌀$, take $A=K_1$. Otherwise, A is finite and every edge of A lies in a triangle. Coxeter word-length parity makes B bipartite and hence triangle-free. Lemma 22 gives an isomorphism of topological groups

$$Aut\left({\mathcal{C}}_{\mathsf{\Gamma }}\right)\cong Aut\left(A\right)\times Aut\left(B\right).$$

If $S\setminus T=⌀$, then $B=K_1$ and $Aut\left(B\right)=\left\{1\right\}$. If $S\setminus T\ne ⌀$ and ${\mathsf{\Gamma }}_0$ is nonflexible, Theorem 7 implies that $Aut\left(B\right)$ is discrete. In either case $Aut\left({\mathcal{C}}_{\mathsf{\Gamma }}\right)$ is discrete. If ${\mathsf{\Gamma }}_0$ is flexible, Theorem 7 implies that $Aut\left(B\right)$ is not compact-by-discrete. Since $Aut\left(B\right)$ is a closed direct factor, Lemma 4 shows that $Aut\left({\mathcal{C}}_{\mathsf{\Gamma }}\right)$ is not compact-by-discrete. Finally, (2) implies (1) because a discrete group is compact-by-discrete. The three conditions are now equivalent. □

Corollary 24.

If $G_{\mathsf{\Gamma }}$ is graphically discrete, then every vertex of T is universal in Γ and ${\mathsf{\Gamma }}_0$ is nonflexible. If every vertex of T is universal in Γ, then

$$G_{\mathsf{\Gamma }}=\left(\prod _{s\in T}{G}_s\right)\times W_{{\mathsf{\Gamma }}_0}$$

is graphically discrete if and only if $W_{{\mathsf{\Gamma }}_0}$ is graphically discrete.

Proof. 

The left action of $G_{\mathsf{\Gamma }}$ on ${\mathcal{C}}_{\mathsf{\Gamma }}$ is geometric. Thus failure of condition (3) in Theorem 23 supplies a witness to failure of graphical discreteness. Under the universal-vertex hypothesis, the displayed splitting holds, and the last assertion follows from Lemma 5. □

For right-angled Coxeter groups, $T=⌀$ and hence ${\mathsf{\Gamma }}_0=\mathsf{\Gamma }$. In this case the flexible obstruction is already supplied by Theorem 7.

3.7. Odd Graphs and Menger Curve Boundaries

For $n\ge 3$, the odd graph

$$O_n=KG(2n-1,n-1)$$

has vertex set consisting of the $(n-1)$-element subsets of $\{1,\dots ,2n-1\}$. Two vertices are adjacent when the corresponding subsets are disjoint [p. 5] [2]. The graph $O_3$ is the Petersen graph.

Proposition 25.

For every $n\ge 3$, the graph $O_n$ is flexible, triangle-free, has no induced four-cycle, is inseparable, and is nonplanar.

Proof. 

Fix a vertex A, so $\left|A\right|=n-1$. A nontrivial permutation of the elements of A, extended by the identity on the complement of A, induces an automorphism of $O_n$. The induced automorphism is nontrivial because it moves an $(n-1)$-subset containing one moved element but not its image. It fixes A and fixes every neighbour of A pointwise, since those neighbours are subsets of the complement. Hence $O_n$ is flexible.

Three pairwise adjacent vertices would be three disjoint $(n-1)$-subsets of a set of size $2n-1$, which is impossible for $n\ge 3$. Thus $O_n$ is triangle-free. If two distinct vertices A and C had two distinct common neighbours, the complement of $A\cup C$ would contain two distinct $(n-1)$-subsets and would therefore have at least n elements. But $|A\cup C|\ge n$, so its complement has at most $n-1$ elements. This contradiction shows that $O_n$ has no four-cycle.

For the Kneser graph $KG(N,r)$ with $N>2r$, every vertex has degree $\left({\displaystyle \genfrac{}{}{0pt}{}{N-r}{r}}\right)$. Its vertex connectivity is also $\left({\displaystyle \genfrac{}{}{0pt}{}{N-r}{r}}\right)$ [Theorem 2] [2]. Thus $O_n$ is n-connected. Dani–Haulmark–Walsh note that a triangle-free graph is inseparable exactly when it is connected and has no separating vertex, separating edge, cut pair, or separating vertex suspension [p. 136] [6]. For $n\ge 4$, deletion of at most three vertices does not disconnect $O_n$, so all four obstructions are absent. For $n=3$, the Petersen graph is 3-connected, and a direct check shows that deleting the three vertices of any induced path of length two leaves a connected graph. The Dani–Haulmark–Walsh criterion gives inseparability.

The Petersen graph is nonplanar. For $n\ge 4$, the graph $O_n$ is n-regular and has girth at least five. If it were planar, Euler’s formula would give

$$|E\left(O_n\right)|\le {\displaystyle \frac{5}{3}}\left(\right|V\left(O_n\right)|-2),$$

whereas regularity gives $|E\left(O_n\right)|=n|V\left(O_n\right)|/2\ge 2|V\left(O_n\right)|$, a contradiction. Hence every $O_n$ is nonplanar. □

Corollary 26.

For every $n\ge 3$, the right-angled Coxeter group $W_{O_n}$ is hyperbolic, its Gromov boundary is homeomorphic to the Menger curve, and it is not graphically discrete.

Proof. 

A right-angled Coxeter group is hyperbolic if and only if its defining graph has no induced four-cycle [Corollary 12.6.3] [7]. Thus $W_{O_n}$ is hyperbolic by Proposition 25. Since $W_{O_n}$ is hyperbolic and $O_n$ is triangle-free, inseparable and nonplanar, Dani–Haulmark–Walsh’s Menger-curve criterion [Corollary 1.7] [6] shows that $\partial W_{O_n}$ is homeomorphic to the Menger curve. Finally, the flexibility of $O_n$ and Theorem 7 show that $W_{O_n}$ is not graphically discrete. □

For these examples with Menger curve boundary, failure of graphical discreteness is already witnessed by the standard Coxeter Cayley graph, without passing to a thick building. They also complement the generic picture suggested by [§1.5, Conjecture 1.9] [16], which predicts that random groups in the few-relator and Gromov density models are graphically discrete asymptotically almost surely as the relator length tends to infinity.

4. Questions

The present arguments do not decide whether the graphically discrete case of Theorem 8 is nonvacuous.

Question 27.

Does there exist a finite thick generalized m-gon with $m\ge 3$ whose incidence graph is nonflexible?

For projective planes, Proposition 18 reduces the $m=3$ case to finding a perspectivity-free plane, while Proposition 20 restricts any such example to square order or to nonsquare order with an odd solvable collineation group. These restrictions do not settle Question 27.

More generally, Corollary 13 rules out graphical discreteness for every $W_{\mathsf{\Gamma }}$ admitting a proper partial conjugation, while Theorem 12 excludes additional groups whenever a one-vertex double is flexible. These obstructions suggest the following group-level problem.

Question 28.

Which right-angled Coxeter groups $W_{\mathsf{\Gamma }}$ are graphically discrete when Γ and every one-vertex double $D_v\mathsf{\Gamma }$ are nonflexible?