2.1 Quasi-median graphs

There exist several equivalent definitions of quasi-median graphs; see, for instance, [Reference Bandelt, Mulder and Wilkeit7]. Below is the definition used in [Reference Genevois28].

The graph $K_{3,2}$ is the bipartite complete graph, corresponding to two squares glued along two adjacent edges; and $K_4^-$ is the complete graph on four vertices minus an edge, corresponding to two triangles glued along an edge. The triangle and quadrangle conditions are illustrated in Figure 1.

Figure 1: Triangle and quadrangle conditions.

It is worth noticing that the gate of x in Y, when it exists, is unique and coincides with the (unique) vertex that minimizes the distance to x in Y. As a consequence, it may be referred to as the projection of x onto Y. Gated subgraphs in quasi-median graphs play the role of convex subcomplexes in CAT(0) cube complexes. We record the following useful criterion for future use; a proof can be found in [Reference Chepoi17] (see also [Reference Genevois28, Proposition 2.6]).

Recall that a clique is a maximal complete subgraph, and that cliques in quasi-median graphs are gated [Reference Bandelt, Mulder and Wilkeit7]. A prism is, roughly speaking, a subgraph which is a product of cliques; more precisely, it is an induced subgraph that decomposes as a Cartesian product of complete graphs such that its cliques are also cliques in the whole graph.

Our lemma requires a few definitions. Given an oriented path $\gamma $ in our graph X, which we decompose as a concatenation of oriented edges $e_1 \cdots e_n$ , one says that $\gamma '$ is obtained from $\gamma $ by:

• • flipping a square, if there exists some $1 \leq i \leq n-1$ such that

  $$ \begin{align*}\gamma'= e_1 \cdots e_{i-1} \cdot a \cdot b \cdot e_{i+2} \cdots e_n,\end{align*} $$
  where $e_i,e_{i+1},b,a$ define an unoriented four-cycle in X;

• • shortening a triangle, if there exists some $1 \leq i \leq n-1$ such that

  $$ \begin{align*}\gamma'= e_1 \cdots e_{i-1} \cdot a \cdot e_{i+2} \cdots e_n,\end{align*} $$
  where $e_i,e_{i+1},a$ define an unoriented three-cycle in X;

• • removing a backtrack, if there exists some $1 \leq i \leq n-1$ such that

  $$ \begin{align*}\gamma'= e_1 \cdots e_{i-1} \cdot e_{i+2} \cdots e_n,\end{align*} $$
  where $e_{i+1}$ is the inverse of $e_i$ .

Lemma 2.2 follows from the simple connectivity of triangle-square complex obtained from X by filling three- and four-cycles (which is an easy consequence of the triangle and quadrangle conditions; see [Reference Brešar, Chalopin, Chepoi, Gologranc and Osajda13, Lemma 5.5] for details) combined either with a standard argument based on disc diagrams or with observation that flipping squares and shortening triangles provide the relations of the fundamental groupoid of X (see [Reference Brown14, Statement 9.1.6] for details).

2.2 Median graphs

A graph X is a median graph if, for all vertices $x,y,z \in X$ , there exists a unique vertex $m \in X$ such that

$$ \begin{align*}\left\{ \begin{array}{l} d(x,y)=d(x,m)+d(m,y), \\ d(x,z)= d(x,m)+d(m,z), \\ d(y,z) = d(y,m)+d(m,z). \end{array} \right.\end{align*} $$

The point m is referred to as the median point of the triple $x,y,z$ . Median graphs are known to define the same objects as CAT(0) cube complexes. Indeed, the one-skeleton of a CAT(0) cube complex is a median graph, and the cube-completion of a median graph, namely the cube complex obtained by filling in all the one-skeleta of cubes in the graph with cubes, is a CAT(0) cube complex. We refer to [Reference Chepoi18] for more information.

2.3 Hyperplanes

Similarly to CAT(0) cube complexes, the notion of hyperplanes is fundamental in the study of quasi-median graphs.

See Figure 2 for examples of hyperplanes in a quasi-median graph. A key observation is that hyperplanes in quasi-median graphs always delimit at least two sectors.

Figure 2: A quasi-median graph and four of its hyperplanes (in orange, red, blue, and green).

We refer to [Reference Genevois28, Section 2.2] (and more particularly to [Reference Genevois28, Proposition 2.30]) for more information about the (fundamental) connection between the geometry of quasi-median graphs and their hyperplanes.

We record the following lemmas for future use.

2.4 Graph products

We conclude our preliminary section by considering graph products of groups and their quasi-median graphs.

Let $\Gamma $ be a simplicial graph, and let $\mathcal {G}= \{ G_u \mid u \in V(\Gamma ) \}$ be a collection of groups indexed by the vertex set $V(\Gamma )$ of $\Gamma $ . The graph product $\Gamma \mathcal {G}$ is defined as the quotient

$$ \begin{align*}\left( \underset{u \in V(\Gamma)}{\ast} G_u \right) / \langle \langle [g,h]=1 \mid g \in G_u, h \in G_v, \{u,v\} \in E(\Gamma) \rangle \rangle,\end{align*} $$

where $E(\Gamma )$ denotes the edge set of $\Gamma $ . The groups in $\mathcal {G}$ are referred to as vertex groups.

2.5 Convention

In the entire article, we will assume for convenience that the groups in $\mathcal {G}$ are nontrivial. Notice that it is not a restrictive assumption, since a graph product with some trivial factors can be described as a graph product over a smaller graph all of whose factors are nontrivial.

A word in $\Gamma \mathcal {G}$ is a product $g_1 \cdots g_n$ where $n \geq 0$ and where, for every $1 \leq i \leq n$ , $g_i \in G$ for some $G \in \mathcal {G}$ ; the $g_i$ are the syllables of the word, and n is the length of the word. Clearly, the following operations on a word does not modify the element of $\Gamma \mathcal {G}$ it represents:

1. Cancellation: delete the syllable $g_i=1$ ;

2. Amalgamation: if $g_i,g_{i+1} \in G$ for some $G \in \mathcal {G}$ , replace the two syllables $g_i$ and $g_{i+1}$ by the single syllable $g_ig_{i+1} \in G$ ;

3. Shuffling: if $g_i$ and $g_{i+1}$ belong to two adjacent vertex groups, switch them.

A word is graphically reduced if its length cannot be shortened by applying these elementary moves. Every element of $\Gamma \mathcal {G}$ can be represented by a graphically reduced word, and this word is unique up to the shuffling operation. This allows us to define the length of an element $g \in \Gamma \mathcal {G}$ , denoted by $|g|$ , as the length of any graphically reduced word representing g. For more information on graphically reduced words, we refer to [Reference Green36] (see also [Reference Genevois32, Reference Hsu and Wise43]).

We record the following definition for future use.

The connection between graph products and quasi-median graphs is made explicit by the following statement [Reference Genevois28, Proposition 8.2].

Notice that the graph product $\Gamma \mathcal {G}$ naturally acts by isometries on ${\mathrm {QM}(\Gamma , \mathcal {G})}$ by left multiplication. We refer to [Reference Genevois28, Section 8.1] (and [Reference Genevois and Martin35, Section 2.2]) for more information about the geometry of ${\mathrm {QM}(\Gamma , \mathcal {G})}$ . Here, we only mention the following two statements, which describe the cliques and the prisms of ${\mathrm {QM}(\Gamma , \mathcal {G})}$ .

3.1 Warm up: special cube complexes revisited

As introduced in [Reference Haglund and Wise40], special cube complexes are nonpositively curved cube complexes which do not contain configurations of hyperplanes referred to as pathological. Then, the key observation is that, given such a cube complex X, there exists a graph $\Gamma $ (namely, the crossing graph of the hyperplanes in X) and a local isometry $X \hookrightarrow X_\Gamma $ , where $X_\Gamma $ is a nonpositively curved cube complex with the right-angled Artin group $A_\Gamma $ as fundamental group (namely, a Salvetti complex). As local isometries between nonpositively curved cube complexes are $\pi _1$ -injective, it follows that $\pi _1(X)$ is a subgroup of $A_\Gamma $ . Similar arguments can be conducted when $A_\Gamma $ is replaced with the right-angled Coxeter group $C_\Gamma $ , but considering either $A_\Gamma $ or $C_\Gamma $ is essentially equivalent because a right-angled Artin groups always embeds as a finite-index subgroup into a right-angled Coxeter group [Reference Davis and Januszkiewicz22].

In this section, we sketch an alternative approach which illustrates the more general arguments from the next section. So, we fix a group G which acts specially on a CAT(0) cube complex X, i.e.,

• • for every hyperplane J and every element $g \in G$ , J and $gJ$ are neither transverse nor tangent;

• • for all hyperplanes $J_1,J_2$ and every element $g \in G$ , if $J_1$ and $J_2$ are transverse, then $J_1$ and $gJ_2$ cannot be tangent.

Observe that this definition is compatible with the terminology introduced by Definition 1.1 from the introduction: because a hyperplane in a CAT(0) cube complex always delimits exactly two subspaces, and the action $\mathfrak {S}(J) \curvearrowright \mathscr {S}(J)$ is automatically free for every hyperplane J, so special and hyperplane-special actions coincide.

Let $\Gamma $ denote the graph whose vertices are the G-orbits of hyperplanes and whose edges link two orbits if they contain at least two transverse hyperplanes. Naturally, each hyperplane of X is labeled by a vertex of $\Gamma $ , namely the G-orbits it belongs to. Define the label of an oriented path $\gamma $ in X as the word $\ell (\gamma )$ given by the sequence of the G-orbits of hyperplanes it crosses. Fixing a basepoint $x_0 \in X$ , we consider

$$ \begin{align*}\Phi : \left\{ \begin{array}{ccc} X & \to & X(\Gamma), \\ x & \mapsto & \ell(\text{path from}\ x_0\ \text{to } x), \end{array} \right.\end{align*} $$

where $X(\Gamma )$ denotes the usual CAT(0) cube complex on which the right-angled Coxeter group $C_\Gamma $ acts, namely the cube completion of the Cayley graph $\mathrm {Cay}(C_\Gamma , V(\Gamma ))$ . Notice that $\Phi $ naturally induces

$$ \begin{align*}\varphi : \left\{ \begin{array}{ccc} G & \to & C_\Gamma, \\ g & \mapsto & \Phi(g \cdot x_0). \end{array} \right.\end{align*} $$

It turns out that $\varphi $ is an injective morphism, that $\Phi $ is a $\varphi $ -equivariant embedding, and that the image of $\Phi $ is a convex subcomplex of $C_\Gamma $ . These observations are based on the following three claims, for which we sketch justifications.

First, consider an oriented path of the form $ee^{-1}$ , namely a backtrack. Then $\ell (ee^{-1})= \ell (e)^2$ equals $1$ in $C_\Gamma $ . Next, consider an oriented path of the form $e \cdot f$ where e and f are consecutive edges in a square. Because the hyperplanes dual to e and f are transverse, the generators $\ell (e)$ and $\ell (f)$ commute in $C_\Gamma $ , so

$$ \begin{align*}\ell(e \cdot f ) = \ell(e) \ell(f) = \ell(f) \ell(e)= \ell(e' \cdot f'),\end{align*} $$

where $e' \cdot f'$ denotes the image of $e \cdot f$ under the diagonal reflection in the square which contains $e \cdot f$ . Therefore, the label of a path remains the same if we add or remove a backtrack or if we flip a square. In a CAT(0) cube complex, any two paths with the same endpoints can be obtained from one to another thanks to such elementary operations, so the desired conclusion follows.▪

Fix two elements $g,h \in G$ . We have

$$ \begin{align*}\varphi(gh) &= \ell([x_0,ghx_0])= \ell ( [x_0,gx_0] \cdot g [x_0,hx_0])\\ &= \ell([x_0,gx_0]) \ell( [x_0,hx_0]) = \varphi(g) \varphi(h),\end{align*} $$

where the second equality is justified by Claim 3.1, and the third one by the fact that the labeling map $\ell $ is G-invariant. (Here, $[\cdot ,\cdot ]$ refers to some arbitrary choice of a geodesic between the two vertices under consideration.)▪

So far, the specialness of the action has not been used, and the morphism $\varphi : G \to C_\Gamma $ is well defined for every action of G on a CAT(0) cube complex. However, this assumption is crucial in the proof of the injectivity of $\Phi $ (and $\varphi $ ), which follows from the next assertion.

Assume that there exists a vertex $x \in X$ and a geodesic $[x_0,x]$ such that the word $\ell ([x_0,x])$ is not graphically reduced. So, if we write $[x_0,x]$ as a concatenation of oriented edges $e_1 \cdots e_n$ , then there exist two indices $1 \leq i < j \leq n$ such that $\ell (e_i)= \ell (e_j)$ and such that $\ell (e_k)$ commutes with $\ell (e_i)$ for every $i < k < j$ . Assume that $j -i \geq 2$ . Because $\ell (e_i)$ and $\ell (e_{i+1})$ commute, the hyperplane $J_i$ dual to $e_i$ has a G-translate which is transverse to the hyperplane $J_{i+1}$ dual to $e_{i+1}$ . Because the action is special, the hyperplanes $J_i$ and $J_{i+1}$ cannot be tangent, so they are transverse. As a consequence, the edges $e_i$ and $e_{i+1}$ span a square, and by flipping this square, we can replace our geodesic $[x_0,x]$ with a new geodesic so that $j-i$ decreases. By iterating the process, we end up with a geodesic $[x_0,x]$ such that $j=i+1$ . In other words, $[x_0,x]$ contains two successive edges with the same label; or equivalently, if J and H denote the hyperplanes dual to these two edges, J and H belong to the same G-orbit. However, J and H are either tangent or transverse, which contradicts the specialness of the action.▪

3.2 Embeddings into graph products

In this section, we define special actions on quasi-median graphs and we show, given a group admitting such an action, how to embed it into a graph product. We begin by introducing the following notation.

Special actions on quasi-median graphs are defined as follows.

It is worth noticing that our definition agrees with the definition of special actions on median graphs we used in the previous section. In other words, an action on a median graph is special if and only if it is hyperplane-special. Indeed, hyperplanes in median graphs delimit exactly two sectors, and a faithful action on a set of cardinality two is automatically free.

As a preliminary observation, note that:

The rest of the section is almost entirely dedicated to the proof of the following embedding theorem.

Let us prove that such objects exist. Fix representatives $S_\alpha $ , $\alpha \in A$ , in $\mathscr {S}(J_i)$ under the action of $\mathfrak {S}(J_i)$ ; and representatives $g_\beta $ , $\beta \in B$ , in $G_i$ under the action by left-multiplication of $\mathfrak {S}(J_i)$ . We know by assumption that the cardinality of B is at least the cardinality of A, so we can assume that B contains A, with equality if they both have the same cardinality. Because $\mathfrak {S}(J_i)$ acts freely on $\mathscr {S}(J_i)$ , every point in $\mathscr {S}(J_i)$ can be uniquely written as $gS_\alpha $ where $g \in \mathfrak {S}(J_i)$ and $\alpha \in A$ . Therefore, setting $\lambda _i (g S_\alpha ) := g g_\alpha $ for all $g \in \mathfrak {S}(J_i)$ and $\alpha \in A$ defines an $\mathfrak {S}(J_i)$ -equivariant injection $\mathscr {S}(S_j) \to G_i$ , which is also surjective if $A=B$ . Of course, we can assume that $S(J_i)$ is one of our representatives, say $S_\gamma $ for some $\gamma \in A$ , and that $g_\gamma =1$ . Then $\lambda _i(S(J_i))=1$ . Thus, our $\lambda _i$ satisfies the desired properties. Next, declare that two sectors $R,S \in \mathscr {S}(J_i)$ have the same color if there exist $g \in \mathfrak {S}(J_i)$ and $\alpha ,\beta \in A$ such that $R=gS_\alpha $ and $S=gS_\beta $ . The third item is satisfied because $\mathfrak {S}(J_i)$ acts freely on $\mathscr {S}(J_i)$ , and the fourth item is satisfied by construction (the uniqueness being also a consequence of the freeness of the action of $\mathfrak {S}(J_i)$ ).

From now on, we fix, for every $i \in I$ , such a map $\lambda _i$ and such a coloring of $\mathscr {S}(J_i)$ . The specific construction does not matter; only the properties recorded by the four items will be used in the sequel. Observe that, in particular, the sectors delimited by $J_i$ are labeled by elements in $G_i$ under $\lambda _i$ . We want to extend such a labeling equivariantly to all the hyperplanes of X.

Of course, there exist $i \in I$ and $h \in G$ such that $hJ=J_i$ . By definition of our colouing of $\mathscr {S}(J_i)$ , there exists some $k \in \mathrm {stab}(J_i)$ such that $k \cdot hS(J)$ and $S(J_i)$ have the same color. Setting $g:=kh$ proves the claim.

3.3 Labeling the sectors

If J is an arbitrary hyperplane of X, let $i \in I$ and $g \in G$ be as given by Claim 3.7. A sector S delimited by J is labeled by $\ell (S):=\lambda _i(gS) \in G_i$ .

Notice that the label of S does not depend on the choice of g. Indeed, let $h \in G$ be another element satisfying Claim 3.7. Then, $gh^{-1}$ stabilizes $J_i$ , and the sectors $gS(J)$ , $S(J_i)$ , and $hS(J)$ all have the same color. In other words, $gh^{-1}$ defines an element of $\mathfrak {S}(J_i)$ that sends some element of $\mathscr {S}(J_i)$ to an element of the same color, which implies that $gh^{-1}$ represents the trivial element of $\mathfrak {S}(J_i)$ . We conclude that $g S=gh^{-1} \cdot hS=hS\!$ .

3.4 Labeling the oriented paths

If $e \subset X$ is an oriented edge, let $S_1$ (resp. $S_2$ ) denote the sector delimited by the hyperplane dual to e which contains the initial endpoint of e (resp. the terminal endpoint of e). The label of e is defined as $\ell (e):= \ell (S_1)^{-1} \ell (S_2)$ . More generally, if $\gamma = e_1 \cdots e_n$ is an oriented path, then its label is defined as the word $\ell (\gamma ):= \ell (e_1) \cdots \ell (e_n)$ , most of the time thought of as an element of $\Gamma \mathcal {G}$ .

Because we may consider the label of an oriented path either as a word or as an element of $\Gamma \mathcal {G}$ , we will use the following notation in order to avoid any ambiguity. Given two labels a and b, we denote by $a=b$ the equality in the group $\Gamma \mathcal {G}$ , and $a \equiv b$ the equality as words.

We record below two fundamental facts about the labeling we have constructed: it is G-invariant, and it sends geodesics to graphically reduced words.

Let J denote the hyperplane dual to e. According to Claim 3.7, there exist $i \in I$ and $h,k \in G$ such that $hJ=J_i=k \cdot gJ$ and such that $S(J_i)$ , $hS(J)$ , and $k S(gJ)$ all have the same color. As a consequence, $kgh^{-1}$ stabilizes $J_i$ , so it defines an element $\sigma $ of $\mathfrak {S}(J_i)$ . Notice that, if S is an arbitrary sector delimited by J, then $\sigma $ sends $hS$ to $k \cdot gS$ (as elements of $\mathscr {S}(J_i)$ ). Hence,

$$ \begin{align*}\ell(gS)=\lambda_i(kgS)=\lambda_i(\sigma hS)= \sigma \lambda_i (hS) = \sigma \ell(S),\end{align*} $$

in $G_i$ . The key observation is that $\sigma $ does not depend on S. Therefore, if $S_1$ (resp. $S_2$ ) denotes the sector delimited by J which contains the initial endpoint of e (resp. the terminal endpoint of e), then

$$ \begin{align*}\ell(ge)= \ell(gS_1)^{-1} \ell(gS_2)= \ell(S_1)^{-1} \sigma^{-1} \sigma \ell(S_2) = \ell(S_1)^{-1} \ell(S_2)= \ell(e),\end{align*} $$

concluding the proof of our claim.

Assume for contradiction that there exist vertices $x,y \in X$ and a geodesic $[x,y]$ from x to y, which we decompose as a concatenation of edges $e_1 \cdots e_r$ , such that $\ell ([x,y])$ is not graphically reduced in $\Gamma \mathcal {G}$ . So, there exist two indices $1 \leq i < j \leq r$ such that $\ell (e_i)$ and $\ell (e_j)$ belong to the same vertex group of $\Gamma \mathcal {G}$ and such that $\ell (e_k)$ belongs to an adjacent vertex group for every $i < k<j$ . In other words, if $J_k$ denotes the hyperplane dual to $e_k$ for every $1 \leq k \leq r$ , then $J_i$ and $J_j$ belong to the same G-orbit and, for every $i<k<j$ , a G-translate of $J_k$ is transverse to $J_j$ . Because G acts specially on X, notice that, if $j\geq i+2$ , then the hyperplane $J_{j-1}$ cannot be tangent to $J_j$ , so $J_{j-1}$ and $J_j$ are transverse. As a consequence of Lemma 2.6, there exists a geodesic from x to y which crosses the hyperplanes $J_1, \ldots , J_{j-2},J_j,J_{j-1},J_{j+1}, \ldots , J_r$ in that order. By iterating the argument, it follows that we can choose carefully our geodesic $[x,y]$ so that $j=i+1$ . In other words, $J_i$ and $J_j$ are tangent or transverse. However, we know that $J_i$ and $J_j$ belong to the same G-orbit, contradicting the specialness of the action. The proof of our claim is complete.

3.5 The embedding

Fix a second basepoint $x_1 \in X$ , possibly different from $x_0$ . In order to prove our theorem, we want to show that

$$ \begin{align*}\Phi : \left\{ \begin{array}{ccc} X & \to & X(\Gamma, \mathcal{G}) \\ x & \mapsto & \ell \left( \text{path from}\ x_1\ \text{to}\ x \right) \end{array} \right.\end{align*} $$

defines an embedding whose image, that

$$ \begin{align*}\varphi : \left\{ \begin{array}{ccc} G & \to & \Gamma \mathcal{G} \\ g & \to & \Phi(g \cdot x_1) \end{array} \right.\end{align*} $$

is an injective morphism, and that $\Phi $ is $\varphi $ -equivariant.

First of all, we claim that $\Phi $ is well defined, i.e., the label of a path from $x_1$ to x (as an element of $\Gamma \mathcal {G}$ ) does not depend on the path we choose. As a consequence of Lemma 2.2, it suffices to show that flipping a square, shortening a triangle, and removing a backtrack do not modify the label of a path.

We begin by noticing that, if $e \cdot f$ is an oriented path between two opposite vertices of a square and if $e' \cdot f'$ denotes the image of $e \cdot f$ under the reflection along the diagonal of our square, then $e \cdot f$ and $e' \cdot f'$ have the same label. Indeed, observe that the endpoints of e and $f'$ belong to the same sectors delimited by the hyperplane dual to e and $f'$ , and similarly for f and $e'$ , so $\ell (e \cdot f) \equiv \ell (e) \ell (f)$ and $\ell (e' \cdot f') \equiv \ell (f) \ell (e)$ . However, $\ell (e)$ and $\ell (f)$ belong to two vertex groups of $\Gamma \mathcal {G}$ which are adjacent since the two hyperplanes dual to e and f are transverse. Therefore,

$$ \begin{align*}\ell(e \cdot f ) =\ell(e) \ell(f)= \ell(f) \ell(e) = \ell(e' \cdot f'),\end{align*} $$

so that flipping a square in a path does not modify its label (in $\Gamma \mathcal {G}$ ). Next, if $e \cdot f$ is a backtrack, then

$$ \begin{align*}\ell(e \cdot f) = \ell(e) \ell(f)= \ell(e) \ell(e)^{-1}= 1,\end{align*} $$

so that removing a backtrack to a path does not modify its label (in $\Gamma \mathcal {G}$ ) either. Finally, let $e \cdot f$ be the concatenation of two successive edges in a triangle, and let $e'$ denote the edge of this triangle with the same endpoints as $e\cdot f$ . Let J denote the hyperplane containing our triangle, $S_1$ the sector delimited by J which contains the initial point of e, $S_2$ the sector delimited by J which contains the terminal endpoint of e, and $S_3$ the sector delimited by J which contains the terminal point of f. Then

$$ \begin{align*}\ell(e \cdot f ) = \ell(e) \ell(f)= \ell(S_1)^{-1} \ell(S_2) \cdot \ell(S_2)^{-1} \ell(S_3)= \ell(S_1)^{-1} \ell(S_3) = \ell(e'),\end{align*} $$

so that shortening a triangle does not modify the label of a path. Thus, we have proved that $\Phi $ is well defined.

It is worth noticing that our map $\Phi $ essentially does not depend on the basepoint $x_1$ we choose. When we allow the basepoint $x_1$ to vary, we denote by $\Phi _z$ the map obtained from $\Phi $ by replacing $x_1$ with another vertex $z \in X$ . Then:

Claim 3.10 For all vertices $p,q \in X$ , we have the commutative diagram

where the isometry $m_g$ denotes the left-multiplication by $g:= \ell ([p,q])$ .

Indeed,

$$ \begin{align*}\Phi_p(x)= \ell([p,x]) = \ell([p,q] \cdot [q,x]) = \ell([p,q]) \cdot \Phi_q(x),\end{align*} $$

for every vertex $x \in X$ .

We are now ready to show that $\varphi $ is an injective morphism and that $\Phi $ is a $\varphi $ -equivariant embedding.

Let $x,y \in X$ be two vertices. Fix a geodesic $[x,y]$ between x and y in X. As a consequence of Claim 3.10,

$$ \begin{align*}d(\Phi(x), \Phi(y))= d( \Phi_x(x), \Phi_x(y)) = d(1, \ell([x,y]).\end{align*} $$

However, $\ell ([x,y]$ is a graphically reduced word according to Claim 3.9, so $d(1, \ell ([x,y]))$ coincides with the length of $\ell ([x,y])$ , or equivalently with the number of edges of $[x,y]$ . We conclude that $d(\Phi (x),\Phi (y))= d(x,y)$ .

By fixing arbitrary paths $[x_1,gx]$ , $[x_1,gx_1]$ , and $[x_1,x]$ in X, we have

$$ \begin{align*}\Phi(gx) = \ell([x_1,gx])= \ell([x_1,gx_1] \cdot g[x_1,x]) = \ell([x_1,gx_1]) \ell([x_1,x])= \varphi(g) \Phi(x),\end{align*} $$

where the penultimate equality is justified by Claim 3.8. Our claim is proved.

Notice that Claim 3.12 implies that $\varphi $ is a morphism. Indeed, for every $g,h \in G$ , we have

$$ \begin{align*}\varphi(gh) = \Phi(gh \cdot x_1)= \varphi(g) \Phi(h \cdot x_1)= \varphi(g) \varphi(h).\end{align*} $$

Moreover, the injectivity of $\varphi $ follows from the injectivity of $\Phi $ provided by Claim 3.11, combined with Lemma 3.5, and Claim 3.12 precisely means that $\Phi $ is $\varphi $ -equivariant.

From now on, we assume that, for every $i \in I$ , the index of $\mathfrak {S}(J_i)$ in $G_i$ coincides with the number of orbits of $\mathfrak {S}(J_i) \curvearrowright \mathscr {S}(J_i)$ ; by construction, this implies that $\lambda _i : \mathscr {S}(J_i) \to G_i$ is a bijection. We want to show that, under this assumption, the image of $\Phi $ is a gated subgraph of ${\mathrm {QM}(\Gamma , \mathcal {G})}$ .

Fix an element $b \in G_i$ . Let C denote the clique of X containing our edge labeled by a, and let J denote the hyperplane of X which contains it. By construction, the sectors delimited by J are labeled by elements of $G_i$ , and conversely every element of $G_i$ labels a sector delimited by J. Let e be the edge of C which connects x to the sector delimited by J which is labeled by $cb$ , where $c \in G_i$ is the label of the sector containing x. Then $\ell (e)= c^{-1} \cdot cb = b$ , so e is the edge we are looking for.

Let C be a clique of X. Fix an arbitrary vertex $x \in C$ . The edges of C are all labeled by the same group $G_i$ , $i \in I$ . It follows from Claim 3.13 that $\Phi (C) \supset \Phi (x) G_i$ . On the other hand, $\Phi (x)G_i$ is a clique in ${\mathrm {QM}(\Gamma , \mathcal {G})}$ according to Lemma 2.8, so $\Phi (C) \subset \Phi (x) G_i$ . This proves the first assertion of our claim. Now, let T be a triangle with at least one edge e in the image of $\Phi $ . Let A be a clique of X such that $\Phi (A)$ contains e, and let B be the clique of $\mathrm {QM}(\Gamma , \mathcal {G})$ that contains T. Then $\Phi (A)$ and B are two cliques of ${\mathrm {QM}(\Gamma , \mathcal {G})}$ containing e. However, in a quasi-median graph, the intersecting between two distinct cliques contains at most one vertex (which is an immediate consequence of the fact that there is no induced $K_4^-$ ); hence, $T \subset B=\Phi (A)$ , proving that the image of $\Phi $ contains its triangle, as desired.

Let $e_1,e_2 \subset X$ be two edges which share their initial point and such that $\Phi (e_1)$ and $\Phi (e_2)$ span a square S. Necessarily, $\ell (e_1)$ and $\ell (e_2)$ belong to adjacent vertex groups, which means that the hyperplane dual to $e_1$ has a G-translate which is transverse to the hyperplane dual to $e_2$ . Because G acts specially on X, it follows that the hyperplanes dual to $e_1$ and $e_2$ are transverse, so that $e_1$ and $e_2$ span a square in X according to Lemma 2.5. The image of this square under $\Phi $ must be S as ${\mathrm {QM}(\Gamma , \mathcal {G})}$ does not contain $K_{3,2}$ as an induced subgraph, concluding the proof of our claim.

By combining Lemma 2.1 with Claims 3.14 and 3.15, we conclude that the image of $\Phi $ is a gated subgraph. The proof of our theorem is complete.

When applying Theorem 3.6, it may be difficult to understand the groups $\mathfrak {S}(J)$ . Our next statement shows that, when the group acts with finitely many orbits of vertices, these groups are essentially clique-stabilizers (which are much easier to understand).

As a consequence of Proposition 3.17, we better understand the vertex groups of the graph product into which we embed our group in Theorem 3.6, under the additional assumption that the action on the quasi-median graph has only finitely many orbits of vertices.

By combining Theorem 3.6 with Corollary 3.19, one immediately gets the following corollary.

3.6 Gated-cocompact subgroups are virtual retracts

We saw in the previous section that a group acting specially on a quasi-median graph can be embedded into a graph product. In the present section, our goal is to show, under the additional assumption that the group acts with only finitely many orbits of vertices, that the image of this embedding is a virtual retract. Our proof is based on the following concept.

Unless stated otherwise, a gated-cocompact subgroup of a graph product $\Gamma \mathcal {G}$ always refers to the action of $\Gamma \mathcal {G}$ on ${\mathrm {QM}(\Gamma , \mathcal {G})}$ . The main result of this section is that such subgroups are virtual retracts.

Before turning to the proof of our theorem, we need to introduce a few definitions. So, let X be a quasi-median graph and G a group acting on it.

• • The rotative-stabilizer of a hyperplane J is $\mathrm {stab}_\circlearrowright (J):= \bigcap \{ \mathrm {stab}(C) \mid {C}\ \subset J\ \text {clique} \}.$

• • Given a G-invariant collection of hyperplanes $\mathcal {J}$ , the action $G \curvearrowright X$ is $\mathcal {J}$ -rotative if, for every $J \in \mathcal {J}$ , the action $\mathrm {stab}_\circlearrowright (J) \curvearrowright \mathscr {S}(J)$ is transitive and free.

• • Given a vertex $x \in X$ , a collection of hyperplanes $\mathcal {J}$ is x-peripheral if there do not exist $J_1,J_2 \in \mathcal {J}$ such that $J_1$ separates x from $J_2$ .

For instance, the action of $\Gamma \mathcal {G}$ on ${\mathrm {QM}(\Gamma , \mathcal {G})}$ is fully rotative [Reference Genevois and Martin35, Proposition 2.21], i.e., it is $\mathcal {J}$ -rotative where $\mathcal {J}$ denotes the collection of all the hyperplanes of ${\mathrm {QM}(\Gamma , \mathcal {G})}$ .

Lemma 3.22 Let G be a group acting on a quasi-median graph X with trivial vertex-stabilizers. Fix a basepoint $x_0 \in X$ , and let $\mathcal {J}$ be an $x_0$ -peripheral collection of hyperplanes. Assume that the action of G on X is $\mathcal {J}$ -rotative. Then

$$ \begin{align*}Y:= \bigcap\limits_{J \in \mathcal{J}} \text{sector delimited by}\ J\ \text{containing } x_0\end{align*} $$

is a fundamental domain for the action of $R:= \langle \mathrm {stab}_\circlearrowright (J) \mid J \in \mathcal {J} \rangle $ on X.

Proof Let $x \in X$ be an arbitrary vertex. Assume that $x \notin , Y$ and let $y \in Y$ denote its projection onto Y (which exists since, as an intersection of gated subgraphs, Y is gated). The last edge of a geodesic $[x,y]$ must be dual to a hyperplane J in $\mathcal {J}$ . Because the action is $\mathcal {J}$ -rotative, there exists some $g \in \mathrm {stab}_\circlearrowright (J)$ which sends x in the sector delimited by J which contains Y. Notice that g sends $[x,y]$ minus its last edge to a path between $gx$ and y, so

$$ \begin{align*}d(gx,Y) \leq d(gx,y) \leq d(x,y)-1.\end{align*} $$

By iterating the argument, we conclude that there exists $r \in R$ such that $rx \in Y$ .

Now, fix an arbitrary vertex $x \in Y$ . For every $J \in \mathcal {J}$ , let $X_J$ denote the union of all the sectors delimited by J which are disjoint from Y. Notice that:

• • If $J_1,J_2 \in \mathcal {J}$ are transverse, then $g_1$ and $g_2$ commute for all $g_1 \in \mathrm {stab}_\circlearrowright (J_1)$ and $g_2 \in \mathrm {stab}_\circlearrowright (J_2)$ [Reference Genevois28, Lemma 8.46] (see also [Reference Genevois and Martin35, Fact 2.22]).

• • If $J_1,J_2 \in \mathcal {J}$ are transverse, then $g \cdot X_{J_2} \subset X_{J_2}$ for every $g \in \mathrm {stab}_\circlearrowright (J_1)$ [Reference Genevois28, Lemma 8.47] (see also [Reference Genevois and Martin35, Proposition 2.21]).

• • If $J_1,J_2 \in \mathcal {J}$ are distinct and not transverse, then $g \cdot X_{J_2} \subset X_{J_1}$ for every $g \in \mathrm {stab}_\circlearrowright (J_1) \backslash \{1\}$ .

• • For every $J \in \mathcal {J}$ and every $g \in \mathrm {stab}_\circlearrowright (J) \backslash \{1\}$ , we have $g \cdot x \in X_J$ .

Therefore, [Reference Genevois28, Proposition 8.44] (see also [Reference Genevois31, Proposition 3.26]) applies, and we deduce from [Reference Genevois28, Fact 8.45] (see also [Reference Genevois31, Fact 3.27]) that $g \cdot x \in \bigcup \limits _{J \in \mathcal {J}} X_J$ for every nontrivial $g \in R$ ; in particular, $g \cdot x \notin Y$ . Thus, we have proved that Y is a fundamental domain for $R \curvearrowright X$ .▪

Proof Let $Y \subset {\mathrm {QM}(\Gamma , \mathcal {G})}$ be a gated subgraph on which H acts with finitely many orbits of vertices. Let $\mathcal {J}$ denote the collection of the hyperplanes of ${\mathrm {QM}(\Gamma , \mathcal {G})}$ which are tangent to Y (i.e., with no edges in Y but whose carriers intersect Y). We set $R:= \langle \mathrm {stab}_\circlearrowright (J) \mid J \in \mathcal {J} \rangle $ and $H^+:= \langle R,H \rangle $ . Notice that $\mathcal {J}$ is H-invariant, so R is a normal subgroup of $H^+$ . Moreover, Y coincides with

$$ \begin{align*}\bigcap\limits_{J \in \mathcal{J}} \text{sector delimited by}\ J\ \text{containing}\ Y,\end{align*} $$

which is a fundamental domain of R according to Lemma 3.22. Therefore, ${H \cap R= \{1\}}$ . It follows that $H^+ = R \rtimes H$ , so that H is a retract in $H^+$ . Moreover, since Y is a fundamental domain of R and because H acts on Y with finitely many orbits of vertices, necessarily $H^+$ acts on ${\mathrm {QM}(\Gamma , \mathcal {G})}$ with finitely many orbits of vertices, which means that $H^+$ is a finite-index subgroup of $\Gamma \mathcal {G}$ . Thus, we have proved that H is a virtual retract in $\Gamma \mathcal {G}$ .▪

According to Theorem 3.6, if a group G acts specially on a quasi-median graph X, then there exists an embedding $\varphi : G \hookrightarrow \Gamma \mathcal {G}$ such that X embeds $\varphi $ -equivariantly into ${\mathrm {QM}(\Gamma , \mathcal {G})}$ as a gated subgraph. As a consequence, if G acts on X with finitely many vertices, then the image of $\varphi $ is a gated-cocompact subgroup of $\Gamma \mathcal {G}$ , so Theorem 3.21 directly implies that:

It also follows from Theorem 3.21 that gated-cocompact subgroups of our group G are virtual retracts in G itself.

As subgraphs in median graphs are gated if and only if they are convex, we recover from Corollary 3.24 that convex-cocompact subgroups in cocompact special groups are virtual retracts [Reference Haglund and Wise40].

4.1 Graphs of groups

We begin this section by fixing the basic definitions and notations related to graphs of groups; essentially, we follow [Reference Serre49]. So far, our graphs were always one-dimensional simplicial complexes, but we need a different definition in order to define graphs of groups. In order to avoid ambiguity, we will refer to the latter as abstract graphs.

Notice that the elements of E are referred to as arrows and not as edges. This terminology will allow us to avoid confusion between arrows of abstract graphs and edges of quasi-median graphs. Below, we define graphs of groups and their associated fundamental groupoids as introduced in [Reference Higgins42].

We record the following definition for future use.

The following normal form, proved in [Reference Higgins42], is central in the quasi-median geometry of right-angled graphs of groups.

Such a word will be referred to as a normal word.

Roughly speaking, we will be interested in graphs of groups gluing graph products. In order to get something interesting for our purpose, we need to control the gluings.

Typically, we glue graph products along “subgraph products” in a canonical way. We refer to Section 4.4 for examples.

In the following, a factor will refer to a vertex group of one of these graph products that label the vertices in our graph of groups. In order to avoid possible confusion, in the sequel vertex groups will only refer to the groups labeling the vertices of the underlying abstract graph of our graph of groups.

Given a right-angled graph of groups $\mathfrak {G}$ and an arrow $e \in E$ , there exists a natural set $T_e$ of left-coset representatives of $\iota _e(G_e)$ in $G_{s(e)}$ : the set of elements of $G_{s(e)}$ represented by graphically reduced words whose tails (see Definition 2.4) do not contain any element of the vertex groups in $\iota _e(G_e)$ . From now on, we fix this choice, and any normal word will refer to this convention.

4.2 Quasi-median geometry

Fix a right-angled graph of groups $\mathfrak {G}$ , and a vertex $\omega \in V$ of its underlying abstract graph. Let $\mathfrak {S} \subset \mathfrak {F}$ denote the union of the arrows of E together with the factors (minus the identity) of the graph products $G_v$ , $v \in V$ . By definition, $\mathfrak {S}$ is a generating set of the fundamental groupoid $\mathfrak {F}$ of $\mathfrak {G}$ .

It is worth noticing that an edge of $\mathfrak {X}$ is naturally labeled either by an arrow of E or by a factor.

Notice that the fundamental group $\mathfrak {F}_{\omega }$ of $\mathfrak {G}$ based at $\omega $ naturally acts by isometries on $\mathfrak {X}$ by left multiplication. Moreover:

We record the following definition for future use.

Notice that, by construction, a leaf is isometric to the Cayley graph of a graph product as given by Theorem 2.7. (See [Reference Genevois28, Lemma 11.11] for more details.)

4.2.1 Path morphisms

Let $\mathfrak {G}$ be a right-angled graph of groups, and let $(V,E, \bar {\cdot },s,t)$ denote its underlying abstract graph. Given an arrow $e \in E$ , we denote by $\varphi _e : \iota _e(G_e) \to \iota _{\bar {e}}(G_e)$ the isomorphism $\iota _{\bar {e}} \circ \iota _e^{-1}$ . A priori, $\varphi _e$ is not defined on $G_{s(e)}$ entirely, but for every subset $S \subset G_{s(e)}$ , we can define $\varphi _e(S)$ as $\varphi _e \left ( S \cap \iota _e(G_e) \right )$ . By extension, if an oriented path $\gamma $ decomposes as a concatenation of arrows $e_1 \cdots e_n$ , we denote by $\varphi _\gamma $ the composition $\varphi _{e_n} \circ \cdots \circ \varphi _{e_1}$ .

Notice that, if G is a factor contained in a vertex group $G_u$ of $\mathfrak {G}$ and if $\gamma $ is a path in the graph of $\mathfrak {G}$ starting from u, then $\varphi _\gamma (G)$ is either trivial (i.e., reduced to $\{1\}$ ) or a factor (different from G in general). Moreover, in the latter case, the equality

$$ \begin{align*}a \cdot e_1 \cdots e_n = e_1 \cdots e_n \cdot \varphi_\gamma (a)\end{align*} $$

holds for every $a \in G$ , where $e_1 \cdots e_n$ is a decomposition of $\gamma $ as a concatenation of arrows.

Given a right-angled graph of groups, a subgroup of automorphisms is naturally associated to each factor.

Definition 4.8 For every factor G contained in a vertex-group $G_u$ of $\mathfrak {G}$ ,

$$ \begin{align*}\Phi(G):= \{ \varphi_c \mid c\ \text{closed path based at}\ u \text{ such that}\ \varphi_c(G)=G \} \leq \mathrm{Aut}(G).\end{align*} $$

These groups of automorphisms are crucial in the study of right-angled graphs of groups. Indeed, as noticed by [Reference Genevois28, Example 11.36], cyclic extensions of an arbitrary group are fundamental groups of right-angled graphs of groups, but we cannot expect to find a geometry common to all the cyclic extensions, so we need additional restrictions on the graphs of groups we look at. As suggested by [Reference Genevois28, Proposition 11.26] and Proposition 4.9, typically we require the $\Phi (G)$ to be trivial, or at least finite.

4.2.3 Hyperplanes

Let $\mathfrak {G}$ be a right-angled graph of groups. The rest of the section is dedicated to the description of the hyperplanes of $\mathfrak {X}$ . It is worth noticing that, as a consequence of [Reference Genevois28, Fact 11.14 and Lemma 11.16], a hyperplane has all its edges labeled either an arrow of $\mathfrak {G}$ or by factors (not a single one in general). In the former case, the hyperplane is of arrow-type; and in the latter case, the hyperplane is of factor-type. Notice that, as a consequence of [Reference Genevois28, Fact 11.14], two hyperplanes of arrow-type cannot be transverse.

Roughly speaking, the carrier of the hyperplane dual to a clique labeled by some factor G is induced by the vertices corresponding to elements of $\mathfrak {F}$ which “commute” with all the elements of G. Because commutation is not well-defined in groupoids, we need to define carefully this idea, which is done by the following definition.

Definition 4.9 Let G be a factor contained in a vertex group $G_v$ of $\mathfrak {G}$ . An element $h \in \mathfrak {F}$ with initial vertex v belongs to the link of G, denoted by $\mathrm {link}(G)$ , if it can be written as a normal word $h_1e_1 \cdots h_ne_nh_{n+1}$ such that:

• • $\varphi _{e_1 \cdots e_{n}}(G)$ is nontrivial;

• • $h_1$ belongs to the subgroup of $G_{v}$ generated by the factors adjacent to G;

• • for every $1 \leq i \leq n$ , $h_{i+1}$ belongs to the subgroup of $G_{t(e_i)}$ generated by the factors adjacent to $\varphi _{e_1 \cdots e_i}(G)$ .

We are now ready to describe the hyperplanes of factor-type of $\mathfrak {X}$ and their stabilizers.

Proposition 4.6 [Reference Genevois28, Proposition 11.21]

Let $C=gG$ be a clique where G is a factor and where $g \in \mathfrak {X}$ . Let J denote the hyperplane dual to C. An edge $e \subset \mathfrak {X}$ is dual to J if and only if $e=g(h_1 \ell ,h_2 \ell )$ for some $h_1,h_2 \in G$ distinct and $\ell \in \mathrm {link}(G)$ . As a consequence, $N(J)= g G \cdot \mathrm {link}(G)$ and the fibers of J are the $gh \cdot \mathrm {link}(G)$ where $h \in G$ .

Corollary 4.7 [Reference Genevois28, Corollary 11.22]

Let $C=gG$ be a clique where G is a factor and where $g \in \mathfrak {X}$ . Let J denote the hyperplane dual to C. Then

$$ \begin{align*}\mathrm{stab}(J)=g \{ k h \mid k \in G, \ h \in \mathrm{link}(G), \varphi_h(G)=G \} g^{-1} .\end{align*} $$

In this statement, $\varphi _h$ is defined as follows. Writing h as a normal word $h_1e_1 \cdots h_ne_nh_{n+1}$ as in Definition 4.9 (this representation being unique according to Proposition 4.1), we refer to $e_1 \cdots e_n$ as the path associated to h. Then, $\varphi _h:= \varphi _{e_1 \cdots e_n}$ . Notice that, by definition of $\mathrm {link}(G)$ , $\varphi _h$ always sends G to another factor, or, in other words, $\varphi _h(G)$ cannot be trivial.

About the hyperplanes of arrow type of $\mathfrak {X}$ , a complete description is not required here. The following statement will be sufficient.

Lemma 4.8 [Reference Genevois28, Lemma 11.24]

Let J be a hyperplane of arrow type in $\mathfrak {X}$ . Then J has exactly two fibers, and they are both stabilized by $\mathrm {stab}(J)$ .

4.3 When is the action special?

In this section, we want to understand when the action of the fundamental group of a right-angled graph of groups on the quasi-median graph constructed in Section 4.2 is special. Our main result in this direction is the following statement.

Observe that, if $(iii)$ does not hold, then we can modify our graph of groups by subdividing the loops and by indexing the new vertices and edges by the edge groups of the corresponding loops. This operation does not modify the fundamental group of the graph of groups, it does not interfere with the conditions $(i)$ , $(ii)$ , and $(iv)$ , and the new graph of groups satisfies $(iii)$ .

We begin by proving the following preliminary lemma.

Proof In order to shorten the notation, we assume that g is trivial. As a consequence of Proposition 4.6, it is clear that $\mathrm {stab}(C)=G$ acts faithfully, freely, and transitively on $\mathscr {S}(J)$ . Therefore, the action $\mathfrak {S}(J) \curvearrowright \mathscr {S}(J)$ is transitive, and it is free if and only if the image of $\mathrm {stab}(C) = G$ in $\mathfrak {S}(J)$ is surjective. However, we know from Proposition 4.6 and Corollary 4.7 that $\mathfrak {S}(J)$ is the set of permutations of $\mathscr {S}(J)= \{h \cdot \mathrm {link}(G) \mid h \in G \}$ induced by

$$ \begin{align*}\mathrm{stab}(J)= \{ km \mid k \in G, m \in \mathrm{link}(G), \varphi_m(G)=G\}.\end{align*} $$

Observe that, for all $km \in \mathrm {stab}(J)$ and $h \cdot \mathrm {link}(G) \in \mathscr {S}(J)$ , we have

$$ \begin{align*}km \cdot \left( h \cdot \mathrm{link}(G) \right)= k \varphi_m^{-1}(h) m \cdot \mathrm{link}(G)= k \varphi_m^{-1}(h) \cdot \mathrm{link}(G),\end{align*} $$

where $\varphi _m \in \Phi (G)$ . If $\Phi (G)$ is trivial, then it is clear that the action of $km$ on $\mathscr {S}(J)$ coincides with the action of $k \in G= \mathrm {stab}(C)$ . Conversely, if the action of $km$ on $\mathscr {S}(J)$ coincides with the action of some $f \in G$ , then $k\varphi _m^{-1}(h) \cdot \mathrm {link}(G)= fh \cdot \mathrm {link}(G)$ for every $h \in G$ . As a consequence of Proposition 4.1 and of the definition of $\mathrm {link}(G)$ , this amounts to saying that $k \varphi _m^{-1}(h) = fh$ for every $h \in G$ . Hence, $k=f$ and $\varphi _m=\mathrm {Id}$ . Because the conclusion holds for every $km \in \mathrm {stab}(J)$ , necessarily $\Phi (G)=\{\mathrm {Id}\}$ .▪

The next observation will be fundamental in our proof.

Proof Write $e=(p,pa)$ and $f=(q,qb)$ where $a \in A$ and $b \in B$ . As a consequence of Proposition 4.6, $f=p(a_1 \ell , a_2 \ell )$ for some distinct $a_1,a_2 \in A$ and some $\ell \in \mathrm {link}(A)$ . We have

$$ \begin{align*}b = q^{-1} \cdot qb = \ell^{-1} a_1^{-1}p^{-1} \cdot pa_2\ell = \varphi_{\ell} \left( a_1^{-1}a_2 \right) = \varphi_\gamma \left( a_1^{-1}a_2 \right),\end{align*} $$

where $\gamma $ is the path in the graph of $\mathfrak {G}$ associated to $\ell $ . Because $\varphi _\gamma $ sends a factor to $\{1\}$ or to another factor, we conclude that $\gamma $ is a path from u to v and that $\varphi _\gamma (A)=B$ , as desired.▪

Now, we are ready to determine when the action of the fundamental group of a right-angled graph of groups on its quasi-median graph is hyperplane-special.

Proof First, assume that the action of the fundamental group of $\mathfrak {G}$ on $\mathfrak {X}$ is not hyperplane-special. There are several cases to consider.

Case 1: There exist a hyperplane J of $\mathfrak {X}$ and an element $g \in \mathfrak {F}_\omega $ such that $gJ$ and J are transverse or tangent.

It is clear that, if there exist two distinct intersecting edges of $\mathfrak {X}$ which are labeled by the same arrow of $\mathfrak {G}$ , then the endpoints of this arrow coincide, so that $(iii)$ does not hold. So, from now on, we assume that J is of factor type. Fix two distinct edges $e_1 \subset J$ and $e_2 \subset gJ$ which share their initial point, and let $A,B$ denote the distinct factors which label them. Notice that, because $e_1$ and $e_2$ intersect, our factors A and B belong to the same vertex group of $\mathfrak {G}$ , say $G_u$ . Because $ge_1$ is labeled by the factor A and is dual to the same hyperplane as $e_2$ , namely $gJ$ , it follows from Lemma 4.11 that there exists in the abstract graph of $\mathfrak {G}$ a closed path c based at u such that $\varphi _c(A)=B$ . In particular, $\varphi _c(A)$ is neither trivial nor A, contradicting $(i)$ .

Case 2: There exist two tangent hyperplanes $J_1,J_2$ of $\mathfrak {X}$ and an element $g\in \mathfrak {F}_\omega $ such that $J_1$ and $g J_2$ are transverse.

We distinguish three cases, depending on whether $J_1$ and $J_2$ are of arrow type or of factor type.

Case 2.1: $J_2$ is of arrow type.

Fix a geodesic $\gamma \subset N(J_1)$ whose initial point belongs to $N(J_1) \cap N(J_2)$ and whose last edge is dual to $gJ_2$ . Crossing $J_2$ corresponds to right-multiplying by the arrow e (or its inverse) which labels $J_2$ . However, such a multiplication is allowed only if the element of the groupoid under consideration has as terminus an endpoint of e (the initial or terminal point of e depending on whether we are multiplying by e or $e^{-1}$ ). Consequently, the initial point of $\gamma $ and one of the last two points of $\gamma $ have the same terminus. According to Lemma 4.3, these two points belong to the same $\mathfrak {F}_\omega $ -orbit. So, there exists some $h \in \mathfrak {F}_\omega $ such that the initial point of $\gamma $ belongs to $N(hgJ_2) \cap N(hJ_1)$ . We already know from Case 1 that, if $J_1$ and $hJ_1$ are tangent or transverse, then $(i)$ cannot hold, so (since their carriers intersect) we suppose that they coincide. Similarly, we suppose that $J_2=hgJ_2$ . As $gJ_2$ and $J_1$ are transverse, it follows that $hgJ_2$ and $hJ_1$ must be transverse as well; but $J_2$ and $J_1$ are tangent, a contradiction.

Case 2.2: $J_1$ and $J_2$ are both of factor type.

Fix two edges $e_1 \subset J_1$ and $e_2 \subset J_2$ which share their initial point, and let $A_1$ and $A_2$ denote the factors which label them, respectively. Notice that, because $e_1$ and $e_2$ intersect, $A_1$ and $A_2$ belong to the same vertex group of $\mathfrak {G}$ , say $G_u$ . Moreover, because $e_1$ and $e_2$ do not span a square, $A_1$ and $A_2$ do not commute in the graph product $G_u$ . Next, fix two edges $f_1 \subset J_1$ and $f_2 \subset gJ_2$ which share their initial endpoint and which span a square, and let $B_1$ and $B_2$ denote the factors which label them, respectively. Notice that, because $f_1$ and $f_2$ intersect, $B_1$ and $B_2$ belong to the same vertex group of $\mathfrak {G}$ , say $G_v$ . Moreover, because $f_1$ and $f_2$ span a square, $B_1$ and $B_2$ commute in the graph product $G_v$ . As $e_1$ and $f_1$ are dual to the same hyperplane, namely $J_1$ , it follows from Lemma 4.11 that there exists a path $\alpha $ in the graph of $\mathfrak {G}$ from u to v such that $\varphi _\alpha (A_1)=B_1$ . Similarly, because $f_2$ and $ge_2$ are dual to $gJ_2$ , there exists a path $\beta $ from u to v such that $\varphi _\beta (A_2)=B_2$ . We conclude that $(ii)$ does not hold.

Case 2.3: $J_1$ is of arrow type and $J_2$ of factor type.

Observe that $gJ_2$ and $gJ_1$ are tangent, that $g^{-1} \cdot gJ_1 = J_1$ is transverse to $gJ_2$ , and that $gJ_2$ is of factor type. Therefore, we conclude thanks to Case 2.1.

Thus, we have proved that, if the conditions $(i)$ – $(iii)$ of our proposition hold, then the action of $\mathfrak {F}_\omega $ on $\mathfrak {X}$ is hyperplane-special. Conversely, assume that one of the conditions $(i)$ – $(iii)$ does not hold.

If $(i)$ does not hold, then there exist a loop c in the graph of $\mathfrak {G}$ based at some vertex u and a factor G in the graph product $G_u$ such that $\varphi _c(G)$ is a factor of $G_u$ distinct from G. Fix an arbitrary vertex $h \in \mathfrak {X}$ whose terminus is u (for instance, a concatenation of arrows from $\omega $ to u). Furthermore, fix a nontrivial element $g \in G$ and write c as a concatenation of arrows $e_1 \cdots e_n$ . Notice that, for every $0 \leq i \leq n$ , we have

$$ \begin{align*}hge_1 \cdots e_i = h e_1 \cdots e_i \varphi_{e_1 \cdots e_i}(g) \ \text{where} \ \varphi_{e_1 \cdots e_i} (g) \neq 1,\end{align*} $$

so $he_1 \cdots e_i$ and $hge_1 \cdots e_i$ are adjacent vertices of $\mathfrak {X}$ . As a consequence, for every $0 \leq i \leq n-1$ , the four vertices $he_1 \cdots e_i$ , $hge_1 \cdots e_i$ , $he_1 \cdots e_{i+1}$ , and $hge_1 \cdots e_{i+1}$ span a square in $\mathfrak {X}$ , so the two edges $(h,hg)$ and $(he_1 \cdots e_n, hge_1 \cdots e_n)$ are dual to the same hyperplane, say J. However, $he_1 \cdots e_n h^{-1} \in \mathfrak {F}_\omega $ sends the edge $(h,hg)$ to the edge $(he_1 \cdots e_n, he_1 \cdots e_ng)$ , and the two edges $(he_1 \cdots e_n, hge_1 \cdots e_n)$ and $(he_1 \cdots e_n, he_1 \cdots e_ng)$ are distinct because

$$ \begin{align*}hge_1 \cdots e_n = h e_1 \cdots e_n \varphi_c(g) \ \text{where}\ \varphi_c(g) \notin G.\end{align*} $$

Therefore, the hyperplanes $he_1 \cdots e_n h^{-1} J$ and J are either tangent or transverse (depending on whether G and $\varphi _c(G)$ commute in $G_u$ ). So, the action of $\mathfrak {F}_\omega $ on $\mathfrak {X}$ is not hyperplane-special.

If $(ii)$ does not hold, then there exist two vertices $u,v$ in the graph of $\mathfrak {G}$ , a path $\alpha $ from u to v, a path $\beta $ from v to u, two commuting factors $A_1,A_2 \subset G_u$ , and two noncommuting factors $B_1,B_2 \subset G_v$ such that $\varphi _\alpha (A_1)=B_1$ and $\varphi _\beta (B_2)=A_2$ . Fix an arbitrary vertex h of $\mathfrak {X}$ whose terminus is v (for instance, a concatenation of arrows from $\omega $ to v) and nontrivial elements $p \in B_1$ , $b \in B_2$ , and $a \in A_1$ . Moreover, write $\alpha $ as the concatenation of arrows $a_1 \cdots a_s$ and $\beta $ as $b_1 \cdots b_r$ . Notice that, for every $0 \leq i \leq r$ , the vertices $hb_1 \cdots b_i$ and $hbb_1 \cdots b_i$ are adjacent as

$$ \begin{align*}hbb_1 \cdots b_i = h b_1 \cdots b_i \varphi_{b_1 \cdots b_i}(b) \ \text{where} \ \varphi_{b_1 \cdots b_i} (b) \neq 1.\end{align*} $$

Consequently, for every $0 \leq i \leq r-1$ , the vertices $hb_1 \cdots b_i$ , $hbb_1 \cdots b_i$ , $hb_1 \cdots b_{i+1}$ , and $hbb_1 \cdots b_{i+1}$ span a square. See Figure 3. Similarly, for every $0 \leq i \leq s$ , the vertices $h \beta a_1 \cdots a_i$ and $h \beta a a_1 \cdots a_i$ are adjacent because

$$ \begin{align*}h \beta a a_1 \cdots a_i = h \beta a_1 \cdots a_i \varphi_{a_1 \cdots a_i}(a) \ \text{where} \ \varphi_{a_1 \cdots a_i}(a) \neq 1;\end{align*} $$

so, for every $0 \leq i \leq s-1$ , the vertices $h\beta a_1 \cdots a_i$ , $h \beta a_1 \cdots a_{i+1}$ , $h \beta a a_1 \cdots a_i$ , and $h \beta a a_1 \cdots a_{i+1}$ span a square. Notice that the edges $(h,hb)$ and $(h,hp)$ do not span a square because $B_2$ and $B_1$ do not commute, so the hyperplane $J_1$ dual to $(h,hb)$ is tangent to the hyperplane $J_2$ dual to $(h,hp)$ . Next, because $A_1$ and $A_2$ commute, we have

$$ \begin{align*}hb \beta a = h \beta \varphi_\beta(b) a = h \beta a \varphi_\beta(b) \ \text{where} \ \varphi_\beta(b) \in A_2 \backslash \{ 1 \},\end{align*} $$

so the vertices $h \beta $ , $hb \beta $ , $h \beta a$ , and $hb \beta a$ span a square. As a consequence, the hyperplane $J_3$ dual to the edge $(h \beta ,h\beta a)$ is transverse to $J_1$ . Finally, observe that $\beta \alpha $ is a loop based at v in the graph of $\mathfrak {G}$ , so $g:= h \beta \alpha h^{-1}$ represents an element of $\mathfrak {F}_\omega $ . Moreover, $g(h,hp)= (h \beta \alpha , h \beta \alpha p)$ belongs to the same clique as the edge $(h \beta \alpha , h \beta a \alpha )$ because

$$ \begin{align*}h \beta a \alpha = h\beta \alpha \varphi_{\alpha}(a) \ \text{and} \ p, \varphi_{\alpha}(a) \in B_1,\end{align*} $$

and hence $J_3=gJ_2$ . Thus, we have proved that $J_1$ and $J_2$ are tangent, but $J_1$ and $gJ_2$ are transverse, showing that the action of $\mathfrak {F}_\omega $ on $\mathfrak {X}$ is not hyperplane-special.▪

Figure 3: Configuration of vertices when $(ii)$ does not hold.

Finally, if $(iii)$ does not hold, then there exists an arrow e which is a loop based at some vertex u of the graph of $\mathfrak {G}$ . Fix an arbitrary vertex h of $\mathfrak {X}$ whose terminus is u (for instance, a concatenation of arrows from $\omega $ to u). Then $h^{-1} e h$ defines an element of $\mathfrak {F}_\omega $ which acts on the bi-infinite line $\{ he^n \mid n \in \mathbb {Z} \} \subset \mathfrak {X}$ as a translation of length 1. Consequently, if J is any hyperplane crossing this line, then J and $h^{-1}eh J$ are tangent, proving that the action of $\mathfrak {F}_\omega $ on $\mathfrak {X}$ is not hyperplane-special.