2.1 Isometric group actions

Let $\Gamma $ be a finitely generated group acting by isometries on the metric space $(X,d_X)$ and let $x\in X$ be an arbitrary base point. The (stable) translation length of this action is the function $\ell _X:\mathbf { conj} \rightarrow {\mathbb {R}}$ given by

$$\begin{align*}\ell_X[g]=\lim_{n\to \infty}{\frac{d_X(g^nx,x)}{n}} \ \text{ for }g\in [g] \text{ in }\mathbf{conj}.\end{align*}$$

The exponential growth rate of this action is the quantity

$$\begin{align*}v_X:=\limsup_{T \to \infty}\frac{\log \left(\#\{g\in \Gamma \colon d_X(gx,x)<T\}\right)}{T}\in [0,+\infty].\end{align*}$$

As for the translation lengths, the exponential growth rate is independent of the chosen base point x. If X is geodesic (or more generally roughly geodesic) and the action of $\Gamma $ on X is proper and cocompact, then $v_X$ is finite. In some cases we can recover the exponential growth rate as the limit

$$\begin{align*}\limsup_{T\to \infty}{\frac{1}{T}\log \# \mathfrak{C}_X(T)}, \end{align*}$$

where each $T>0$ we denote

$$\begin{align*}\mathfrak{C}_X(T)=\{[g]\in \mathbf{conj} \colon \ell_X[g]<T\}.\end{align*}$$

This happens for example when $\Gamma $ is hyperbolic and the action on X is proper and cocompact.

Given two isometric actions of $\Gamma $ on the metric spaces X and $X_\ast $ , the Manhattan curve for the pair $(X, X_\ast )$ is the boundary of the convex set

$$\begin{align*}\mathcal{C}_{X_\ast/X}:=\left\{ (a,b) \in {\mathbb{R}}^2 : \sum_{[g] \in \mathbf{conj}} e^{-a\ell_{X_\ast}[g] - b\ell_{X}[g]} < \infty \right\}, \end{align*}$$

assuming it is nonempty. Equivalently, $\mathcal {C}_{X_\ast /X}$ is the set of points $(s,\theta _{X_\ast /X}(s))$ where $\theta _{X_\ast /{X}}(s)$ is the abscissa of convergence of the series

$$\begin{align*}t \mapsto \sum_{[g] \in \mathbf{conj}} e^{-t\ell_{X}[g] - s\ell_{X_\ast}[g]}. \end{align*}$$

By abuse of notation, $\theta _{X_\ast /X}$ is also called the Manhattan curve for $(X,X_\ast )$ .

2.2 $\mathrm {CAT}(0)$ cube complexes

For bibliography about $\mathrm {CAT}(0)$ cube complexes and groups acting on them, we refer the reader to [Reference Bridson and Haefliger10, Reference Sageev74]. A nonpositively curved (NPC) cube complex is a metric polyhedral complex in which all polyhedra are unit-length Euclidean cubes, and satisfies Gromov’s link condition: the link of each vertex is a flag complex. If this complex is simply connected we say that it is a $\mathrm {CAT}(0)$ cube complex.

Let $\mathcal {X}$ be an NPC cube complex. Consider the minimal equivalence relations on the set of edges (resp. oriented edges) of $\mathcal {X}$ such that the edges $e=\{v,w\}$ and $e=\{v',w'\}$ (resp. oriented edges $v\xrightarrow {e} w$ and $v'\xrightarrow {e'} w'$ ) are in the same equivalence class if $v,w,v',w'$ span a square in $\mathcal {X}$ (resp. $v,w,v'w'$ span a square in $\mathcal {X}$ with $v,v'$ adjacent and $w,w'$ adjacent). Equivalence classes of these equivalence relations are called hyperplanes (resp. oriented hyperplanes), and we let $\mathbb {H}(\mathcal {X})$ denote the set of hyperplanes of $\mathcal {X}$ . If the equivalence class of an (oriented) edge e is the (oriented) hyperplane $\mathfrak {h}$ , then we say that e dual to $\mathfrak {h}$ . A hyperplane is 2-sided if it corresponds to exactly two oriented hyperplanes; otherwise it is 1-sided.

Suppose now that $\mathcal {X}$ is a $\mathrm {CAT}(0)$ cube complex, in which case all hyperplanes are 2-sided. A combinatorial path in $\mathcal {X}$ is a sequence $\gamma =(\gamma _0,\dots ,\gamma _n)$ of vertices in $\mathcal {X}$ such that $\gamma _i$ is adjacent to $\gamma _{i+1}$ for $i=0,\dots ,n-1$ . In that case, we say that $\gamma _0$ is the initial vertex of $\gamma $ and $\gamma _n$ is its final vertex. The length of $\gamma =(\gamma _0,\dots ,\gamma _n)$ is defined as n. This path is often seen as a continuous path by also considering the edges $e_{i+1}$ joining each $\gamma _i$ with $\gamma _{i+1}$ . Such a path is geodesic if no two distinct edges $e_i$ are dual to the same hyperplane. The combinatorial metric on $\mathcal {X}$ is the graph metric $d_{\mathcal {X}}$ on its 1-skeleton $\mathcal {X}^1$ so that each edge has length 1. It follows that a combinatorial path is geodesic if and only if it is geodesic for the metric $d_{\mathcal {X}}$ .

A hyperplane $\mathfrak {h}$ in the $\mathrm {CAT}(0)$ cube complex $\mathcal {X}$ separates two vertices of $\mathcal {X}$ if some (any) combinatorial path connecting these vertices has an edge dual to $\mathfrak {h}$ . It follows that the combinatorial distance of any two vertices in $\mathcal {X}$ equals the number of hyperplanes separating them. Also, a hyperplane $\mathfrak {h}$ determines the equivalence relation of “not being separated by $\mathfrak {h}$ ” on the set of vertices of $\mathcal {X}$ . This equivalence relation has exactly 2 equivalence classes $\{\mathfrak {h}^-,\mathfrak {h}^+\}$ , which are the halfspaces determined by $\mathfrak {h}$ . A subcomplex of $\mathcal {X}$ is convex if its vertex set is the intersection of halfspaces. Equivalently, $Z\subset \mathcal {X}$ is convex if any combinatorial geodesic joining points in $Z^0$ is contained in $Z^0$ .

An orthotope structure on the $\mathrm {CAT}(0)$ cube complex $\mathcal {X}$ is a function

$$ \begin{align*}\mathfrak{w}: \mathbb{H}(\mathcal{X}) \rightarrow {\mathbb{R}}_{>0},\end{align*} $$

and the pair $\mathcal {X}^{\mathfrak {w}}=(\mathcal {X},\mathfrak {w})$ is called a cuboid complex. An orthotope structure induces a metric $d_{\mathcal {X}}^{\mathfrak {w}}$ on $\mathcal {X}^1$ by declaring each edge e to have length $\mathfrak {w}(\mathfrak {h})$ for $\mathfrak {h}$ the hyperplane dual to e. In this way, for any two vertices $x,y\in \mathcal {X}^0$ we have

$$\begin{align*}d_{\mathcal{X}}^{\mathfrak{w}}(x,y)=\sum_{\mathfrak{h}\in \mathbb{H}(x|y)}{\mathfrak{w}(\mathfrak{h})},\end{align*}$$

where $\mathbb {H}(x|y)\subset \mathbb {H}(\mathcal {X})$ is the collection of hyperplanes separating x and y. Note that if $\mathfrak {w}$ is the constant function equal to 1, then $d_{\mathcal {X}}^{\mathfrak {w}}$ is just the standard combinatorial metric $d_{\mathcal {X}}$ .

Now let $\Gamma $ be a group acting on the $\mathrm {CAT}(0)$ cube complex $\mathcal {X}$ . We always assume that the action is cubical, meaning that it preserves the cube complex structure. Under this assumption, the action is isometric on $\mathcal {X}^1$ with the combinatorial metric $d_{\mathcal {X}}$ . Similarly, $\ell _{\mathcal {X}}$ always denotes the stable translation length of $\Gamma $ with respect to the action on $(\mathcal {X}^1,d_{\mathcal {X}})$ . If the action of $\Gamma $ is proper and cocompact, we say that $\mathcal {X}$ is a cubulation of $\Gamma $ .

The action of $\Gamma $ on $\mathcal {X}$ induces a natural action on $\mathbb {H}(\mathcal {X})$ . If $\mathfrak {w}$ is a $\Gamma $ -invariant orthotope structure on $\mathcal {X}$ in the sense that $\mathfrak {w}(\mathfrak {h})=\mathfrak {w}(g\mathfrak {h})$ for all $g\in \Gamma $ and $\mathfrak {h}\in \mathbb {H}(\mathcal {X})$ , then we say that $\Gamma $ acts on the cuboid complex $(\mathcal {X},\mathfrak {w})$ . In that case the action of $\Gamma $ on $(\mathcal {X}^1,d_{\mathcal {X}}^{\mathfrak {w}})$ is also by isometries. We let $\ell _{\mathcal {X}}^{\mathfrak {w}}$ denote the stable translation length of $\Gamma $ for its action on $(\mathcal {X}^1,d_{\mathcal {X}}^{\mathfrak {w}})$ .

By a hyperplane stabilizer we mean a subgroup of $\Gamma $ consisting of group elements g such that $g \mathfrak {h}=\mathfrak {h}$ for some fixed hyperplane $\mathfrak {h}\in \mathbb {H}(\mathcal {X})$ . We denote the hyperplane stabilizer of $\mathfrak {h}$ by $\Gamma _{\mathfrak {h}}$ . A hyperplane $\mathfrak {h}$ is essential for the action of $\Gamma $ if for any vertex x of $\mathcal {X}$ , the halfspaces $\mathfrak {h}^\pm $ contain elements in the $\Gamma $ -orbit of x arbitrarily far from $\mathfrak {h}^{\mp }$ . The action of $\Gamma $ on $\mathcal {X}$ is essential if every hyperplane is essential.

If $\mathcal {X}$ is a cubulation of $\Gamma $ , a subgroup $H<\Gamma $ is called convex-cocompact with respect to $\mathcal {X}$ if there exists a convex subcomplex $Z\subset \mathcal {X}$ that is H-invariant and so that the action of H on Z is cocompact. Such a subcomplex Z is called a convex core for H. Note that the hyperplane stabilizer of any hyperplane $\mathfrak {h}$ is convex-cocompact since it acts cocompactly on the (convex subcomplex spanned by the) set of vertices in edges dual to $\mathfrak {h}$ [Reference Haglund and Wise47, Lemma 13.4].

If $\mathcal {X}$ is a $\mathrm {CAT}(0)$ cube complex and $\mathbb {W}\subset \mathbb {H}(\mathcal {X})$ is any collection of hyperplanes, in [Reference Caprace and Sageev20, Section 2.3] Caprace and Sageev introduced the restriction quotient, which is a $\mathrm {CAT}(0)$ cube complex $\mathcal {X}(\mathbb {W})$ equipped with a surjective cellular map $\phi :\mathcal {X} \rightarrow \mathcal {X}(\mathbb {W})$ satisfying the following: an edge in $\mathcal {X}$ is collapsed to a single vertex under $\phi $ if and only if it is dual to a hyperplane not in $\mathbb {W}$ . The projection $\phi $ induces a natural bijection between $\mathbb {W}$ and $\mathbb {H}(\mathcal {X}(\mathbb {W}))$ , and hence any orthotope structure $\mathfrak {w}$ on $\mathcal {X}$ induces an orthotope structure $\phi _\ast (\mathfrak {w})$ on $\mathcal {X}(\mathbb {W})$ .

A cubulation $\mathcal {X}$ of $\Gamma $ is co-special if the quotient $\overline {\mathcal {X}}=\Gamma \backslash \mathcal {X}$ is a special cube complex in the sense of Haglund-Wise [Reference Haglund and Wise47]. Equivalently, $\mathcal {X}$ is co-special if $\Gamma $ injects into a right-angled Artin group $A_G$ inducing a $\Gamma $ -equivariant isometric embedding of $\mathcal {X}$ into $R_G$ as a convex subcomplex, where $R_G$ is the universal cover of the Salvetti complex $\overline {R}_G$ associated to the graph G. Among other properties of special cube complexes, hyperplanes are 2-sided and embedded, and they do not self-osculate [Reference Haglund and Wise47, Def. 3.2]. In particular, different oriented edges with the same initial vertex are dual to different oriented hyperplanes. The cubulation $\mathcal {X}$ of $\Gamma $ is virtually co-special if there exists a finite-index subgroup $\overline {\Gamma } <\Gamma $ such that the action of $\overline {\Gamma }$ on $\mathcal {X}$ is co-special.

Fundamental groups of compact special cube complexes are residually finite, and more generally, their convex-cocompact subgroups are separable [Reference Haglund and Wise47, Corollary 7.9]. For our purposes, co-special cubulations will be used to construct finite-state automata parameterizing combinatorial geodesics, as we explain in Example 2.5 in the next subsection.

2.3 Finite-state automata

For references on automatic structures and some of their connections with group theory, see [Reference Epstein, Cannon, Holt, Levy, Paterson and Thurston32]. For the relation between automatic structures and special cube complexes, we refer the reader to [Reference Li and Wise58, Section 5]. Let S be a finite set and let $S^\ast $ denote the set of finite words over the alphabet S. If $w=h_1\cdots h_n$ and $w'=h^{\prime }_1\cdots h_m'$ are words in $S^\ast $ , then its concatenation is the word $ww':=h_1\cdots h_nh^{\prime }_1\cdots h_m'$ . The length of a word is the number of letters in S composing it. We let the empty set correspond to the unique word of length 0 in $S^\ast $ . A language over S is any subset of words in $S^\ast $ .

A (finite-state) automaton over S is a tuple

$$\begin{align*}\mathcal{A}=(\mathcal{G},\pi,I,F),\end{align*}$$

where $\mathcal {G}=(V,E)$ is a finite directed graph, $\pi :E\rightarrow S$ is a labeling function and $I,F\subset V$ are nonempty sets of initial and final states.

By a path in $\mathcal {G}$ we mean a sequence $\omega $ of (always directed) edges $e_1,\dots ,e_n$ in E such that the final vertex $v_i$ of $e_i$ is the initial vertex of $e_{i+1}$ for $i=1,\dots , n-1$ . If $v_0$ is the initial vertex of $e_1$ , we denote this path $\omega $ either by $\omega =(\xrightarrow { e_1} \cdots \xrightarrow {e_n})$ or $\omega =(v_0\xrightarrow { e_1} \cdots \xrightarrow {e_n} v_n)$ depending on the emphasis we want to give to the vertices. If there is no ambiguity on the edges, we can also denote this path by $(v_0 \rightarrow \cdots \rightarrow v_n)$ . The length of a path is the number of edges that determine it. Note that paths of length 1 correspond to the edges in E. We also allow the degenerate case of paths of length 0, which are the vertices in V.

If $\omega , \omega '$ are paths in $\mathcal {G}$ such that the final vertex of $\omega $ is the initial vertex of $\omega '$ , then the concatenation $\omega \omega '$ is the path in $\mathcal {G}$ defined in the expected way. Similarly we define the concatenation of any finite number of paths.

A word w in $S^\ast $ is represented by the path $\omega =(v_0\xrightarrow { e_1} v_1\cdots \xrightarrow {e_n} v_n)$ in $\mathcal {G}$ if $w=\pi (\omega ):=\pi (e_1)\cdots \pi (e_n)$ . If in addition $v_0\in I$ and $v_n\in F$ , we say that w accepted by $\mathcal {A}$ and that $\omega $ is admissible. It is clear that the word represented by a concatenation $\omega \omega '$ is the concatenation of the words represented by $\omega $ and $\omega '$ . Let $L=L_{\mathcal {A}}$ be the language consisting of the words accepted by $\mathcal {A}$ . In this case we say that L is parametrized by $\mathcal {A}$ .

The automaton $\mathcal {A}$ is deterministic if any two distinct edges in $\mathcal {G}$ with the same initial vertex have different labels. In that case, for any $w\in L_{\mathcal {A}}$ and any initial state $v\in I$ there exists at most one path in $\mathcal {G}$ representing w and starting at v. The automaton is pruned if any vertex in $\mathcal {G}$ is the final vertex of a path starting at an initial state.

Example 2.4 (Automatic structures on hyperbolic groups)

Let $\Gamma $ be a hyperbolic group and consider a finite set $S\subset \Gamma $ generating $\Gamma $ as a semi-group and with word length $|\cdot |_S$ . We consider S as our alphabet, so that there is a natural evaluation map $\text {ev}:S^\ast \rightarrow \Gamma $ . For a fixed total order on S we obtain the lexicographic order $\prec $ on $S^*$ . Let $L=L_S$ be language of lexicographically first geodesics. That is, for each $g \in \Gamma $ , a word $w \in S^\ast $ with $\text {ev}(w)=g$ is in L if and only if w has length $|g|_S$ and $w \prec w'$ for all other $w' \in S^\ast $ with $\text {ev}(w')=g$ and of length $|g|_S$ . Cannon [Reference Cannon15] showed that the language L defined above is regular, in the sense that $L=L_{\mathcal {A}}$ for $\mathcal {A}=(\mathcal {G}=(V,E),\pi ,\{\ast \},V)$ a deterministic finite-state automaton over S. In particular, the evaluation map gives us a length-preserving bijection from L onto $\Gamma $ .

Consider now another finite generating subset $S_\ast \subset \Gamma $ . In [Reference Calegari and Fujiwara14, Lemma 3.8], Calegari and Fujiwara constructed a new deterministic automaton $\mathcal {A}'=(\mathcal {G}'=(V',E'),\pi ',\{\ast '\},V')$ over S parameterizing $L_S$ and an integer-valued function $\phi :V' \rightarrow {\mathbb {Z}}$ such that for any word $w\in L_S$ represented by the path $\omega =(\ast '\xrightarrow { e_1} \cdots \xrightarrow {e_n} v_n)$ in $\mathcal {G}'$ we have

$$\begin{align*}|\text{ev}(w)|_{S_\ast}=\sum_{i=1}^{n}\phi(v_i).\end{align*}$$

We note that it is equivalent to define the labeling $\phi $ on the directed edge set $E'$ instead.

We will see in the following subsection that the automatic structure discussed above gives rise to a dynamical system $(\Sigma ,\sigma )$ called a subshift of finite type. The labeling $\phi $ above corresponds to a real-valued function on $\Sigma $ which satisfies the Markovian property of being ‘constant on $2$ -cylinders’: see Subsection 2.4.

Example 2.5 (Automatic structures on special cube complexes)

For this example we follow Sections 5.2 and 5.3 in [Reference Li and Wise58]. Let $\overline {\mathcal {Z}}$ be a compact special cube complex with universal cover $\mathcal {Z}$ and fix a base vertex $o \in \mathcal {Z}$ . Let $S_{\overline {\mathcal {Z}}}$ be the set of the oriented hyperplanes of $\overline {\mathcal {Z}}$ . In [Reference Li and Wise58], Li and Wise constructed a deterministic pruned finite-state automaton

$$\begin{align*}\mathcal{A}_{\overline{\mathcal{Z}}}=(\mathcal{G}_{\overline{\mathcal{Z}}}=(V,E),\pi,\{\ast\},V)\end{align*}$$

over $S_{\overline {\mathcal {Z}}}$ and parameterizing a language $L_{\overline {\mathcal {Z}}}$ . This language describes geodesics in $\mathcal {Z}$ based at o in the following sense. Given any word $w \in L_{\overline {\mathcal {Z}}}$ represented by the (necessarily unique) admissible path $\omega $ there exists a geodesic path $\gamma _{w}$ in $\mathcal {Z}$ starting at o and ending at the vertex $\tau _{\mathcal {Z}}(w)$ such that:

• ○ if $w=\mathfrak {h}_1\cdots \mathfrak {h}_n\in L_{\overline {\mathcal {Z}}}$ and $\omega =(\ast '\xrightarrow { e_1}\cdots \xrightarrow {e_n} v_n)$ , then $\gamma _w=(o=x_0,\dots ,x_n)$ is a geodesic of length n;

• ○ for each $1 \leq i \leq n$ we have $\pi (e_i)=\mathfrak {h}_i$ and the oriented hyperplane in $\mathcal {Z}$ dual to the oriented edge from $x_{i-1}$ to $x_{i}$ maps to $\mathfrak {h}_i$ under the quotient $\mathcal {Z} \rightarrow \overline {\mathcal {Z}}$ ; and,

• ○ the map $\tau _{\mathcal {Z}}: L_{\overline {\mathcal {Z}}} \rightarrow \mathcal {Z}^0$ is a bijection.

The automaton $\mathcal {A}_{\overline {\mathcal {Z}}}$ being deterministic implies that the geodesic $\gamma _w$ is uniquely determined by w.

Remark 2.6. The language constructed in [Reference Li and Wise58] actually depends on an injection of $\Gamma =\pi _1(\overline {\mathcal {Z}})$ into a right-angled Artin group $A_G$ inducing a $\Gamma $ -equivariant isometric embedding of $\mathcal {Z}$ into $R_G$ as a convex subcomplex, where $R_G$ is the universal cover of the Salvetti complex $\overline {R}_G$ associated to G. In that case, the language obtained is over the alphabet of oriented hyperplanes of $\overline {R}_G$ . The language $L_{\overline {\mathcal {Z}}}$ described above is a particular case of this construction, when we consider the local isometric immersion $\overline {\mathcal {Z}} \rightarrow \overline {R}_{G_{\overline {\mathcal {Z}}}}$ for $G_{\overline {\mathcal {Z}}}$ being the crossing graph of $\overline {\mathcal {Z}}$ . For this immersion there is a natural bijection between $S_{\overline {\mathcal {Z}}}$ and the set of oriented hyperplanes in $\overline {R}_{G_{\overline {\mathcal {Z}}}}$ , see for instance [Reference Haglund and Wise47, Lemma 4.1].

2.4 Symbolic dynamics

In this subsection we introduce the preliminary material we need from symbolic dynamics. See Chapter $1$ of [Reference Parry and Pollicott66] for more information regarding the basic definitions we now present. Let A be a $k \times k$ matrix with entries $0$ or $1$ . This matrix is said to be aperiodic if there exists $N \ge 1$ such that all of the entries of $A^N$ are strictly positive. We say that A is irreducible if for any $i,j \in \{1, \ldots , k \}$ there exists $n \ge 1$ such that $(A^n)_{i,j}$ (i.e., the $(i,j)$ th entry of $A^n$ ) is strictly positive.

The (one-sided) subshift of finite type $\Sigma _A$ associated to A is the set of infinite sequences

$$\begin{align*}\Sigma_A = \left\{(x_n)_{n=0}^{\infty} : x_n \in \{1, \ldots, k\} \text{ and } A_{x_n, x_{n+1}} =1 \text{ for all } n \ge 0\right\}. \end{align*}$$

These infinite sequences can be seen as infinite paths in a directed graph $\mathcal {G}_A$ with vertices labeled $1, \ldots , k$ and a directed edge from vertex i to j if and only if $A_{i,j} =1$ . We will therefore refer to the numbers $1, \ldots , k$ as the states of $\Sigma _A$ . We equip $\Sigma _A$ with the shift map $\sigma : \Sigma _A \to \Sigma _A$ defined by

$$\begin{align*}\sigma((x_n)_{n=0}^\infty) = (x_{n+1})_{n=0}^\infty \end{align*}$$

to obtain a dynamical system $(\Sigma _A, \sigma )$ .

Consider a finite ordered string $x_0, \ldots , x_{m-1} \in \{1, \ldots , k \}$ where $A_{x_j, x_{j+1}} =1$ for each $j=0, \ldots , m-2$ . The cylinder set associated to this string is the subset of $\Sigma _A$ given by

$$\begin{align*}[x_0, \ldots, x_{m-1}] := \left\{ (y_n)_{n=0}^\infty \in \Sigma_A : y_j = x_j \text{ for } j=0, \ldots, m-1 \right\}. \end{align*}$$

We endow $\Sigma _A$ with a topology by declaring the set of all cylinder sets to be an open basis.

The system $(\Sigma _A, \sigma )$ is said to be mixing if for any two open sets $U, V \subset \Sigma _A$ there is $N \ge 1$ such that $\sigma ^n(U) \cap V \neq \emptyset $ for all $n \ge N$ . We say that $(\Sigma _A, \sigma )$ is transitive if for any two open sets $U,V \subset \Sigma _A$ there exists $n \ge 1$ such that $\sigma ^n(U) \cap V \neq \emptyset .$ We have that $(\Sigma _A, \sigma )$ is mixing if and only if A aperiodic and $(\Sigma _A, \sigma )$ is transitive if and only if A is irreducible. We will often suppress the dependence of A in the notation for a subshift and will write $(\Sigma , \sigma )$ .

Throughout the rest of the section $(\Sigma , \sigma )$ will be a mixing subshift of finite type, and consider a function (which we will, at some points, refer to as a potential) $\psi :\Sigma \to {\mathbb {R}}$ . We say that $\psi $ is constant on $2$ -cylinders if $\psi $ is constant on each set of the form $[x_0, x_1]$ where $x_0, x_1 \in \{1, \ldots , k \}$ and $A_{x_0, x_1} = 1$ . The assumption that $\psi $ is constant on $2$ -cylinders guarantees that $\psi $ has Markovian behaviour: the value that $\psi $ takes at $x \in \Sigma $ depends only on the initial cylinder that x belongs to. That is, $\psi (x)$ does not depend on the future cylinders that x visits under the iterates of $\sigma $ . Although this is a restrictive condition for a general function on $\Sigma $ , it is not restrictive for our purposes. This is because we are interested in understanding the growth rate properties of functions that have lattice image: their image lies in $\alpha \mathbb {Z}$ for some $\alpha \in \mathbb {R}$ (as discussed in Subsection 1.3). Hölder continuous functions on $\Sigma $ that have a lattice image have the property that their image only depends on finitely many symbols of the input. Then after relabeling the subshift $\Sigma $ (i.e., moving to a topologically conjugate subshift) we can assume that the function is in fact constant on $2$ -cylinders. See for example the proof of Proposition 5.1 in [Reference Parry and Pollicott66] for an example of this argument. To summarize, functions that are constant on $2$ -cylinders naturally arise when studying discrete geometries. Lastly, it is worth mentioning that such functions have particularly nice properties: see for example Lemma 3.4 below.

For each $n\ge 1$ , the nth Birkhoff sum of $\psi $ is the function

$$\begin{align*}\psi^n : \Sigma \to {\mathbb{R}} \ \text{ such that } \ \psi^n(x) := \psi(x) + \psi(\sigma(x)) + \cdots + \psi(\sigma^{n-1}(x)). \end{align*}$$

A point $x \in \Sigma $ is said to be periodic if $\sigma ^n(x) = x$ for some $n \ge 1$ . Such an n is called a period of x. Note that a periodic point has infinitely many periods. Given a periodic point we will assume that it comes with with a choice of period (which may not be its least period) which we will label $|x|$ (so that $\sigma ^{|x|}(x) = x$ ).

Example 2.8. Consider the automaton $\mathcal {A}'=(\mathcal {G}'=(V',E'),\pi ',\{\ast '\},V')$ and the integer-valued function $\phi :V' \rightarrow {\mathbb {Z}}$ introduced in Example 2.4. Then, as discussed in Example 2.7, $\mathcal {A}'$ gives rise to a subshift of finite type $(\Sigma ,\sigma ).$ Furthermore, the labeling $\phi $ defines a function $f: \Sigma \to {\mathbb {Z}}$ by

$$\begin{align*}f(x) = \phi(v_{x_0}), \end{align*}$$

where $x = (x_n)_{n=0}^\infty $ and $v_{x_0} \in V$ is the vertex corresponding to the symbol $x_0$ . The function f is constant on $2$ -cylinders and furthermore if $x = (x_n)_{n=0}^\infty \in \Sigma $ then

$$\begin{align*}f^n(x) = |g_{x,n}|_{S_\ast} \end{align*}$$

where $g_{x,n} \in \Gamma $ is the group element obtained from multiplying the first n labelings in the infinite path corresponding to x, that is, if the first n edges in the path corresponding to x are $e_1,\ldots , e_n$ then $f^n(x) = |\pi (e_1)\cdots \pi (e_n)|_{S_\ast }$ . We note that, when $x \in \Sigma $ satisfies that $\sigma ^n(x) = x$ then

$$\begin{align*}mf^n(x) = f^{mn}(x) = |g_{x,mn}|_{S_\ast} = |g_{x,n}^m|_{S_\ast} \end{align*}$$

and so

$$\begin{align*}f^n(x) = \lim_{m\to\infty} |g_{x,n}^m|_{S_\ast} / m = \ell_{S_\ast}[g_{x,n}]. \end{align*}$$

Two functions $\psi , \varphi : \Sigma \to {\mathbb {R}}$ , which we are assuming to be constant on $2$ -cylinders, are said to be cohomologous if there exists a continuous function $u: \Sigma \to {\mathbb {R}}$ such that $\psi (x) = \varphi (x) + u(\sigma (x)) - u(x)$ for all $x \in \Sigma $ . By Livsic’s Theorem [Reference Parry and Pollicott66, Proposition 3.7], $\psi $ and $\varphi $ are cohomologous if and only if $\psi ^n(x) = \varphi ^n(x)$ whenever $\sigma ^n(x) = x.$

The variational principle states that there is a unique $\sigma $ -invariant Borel probability measure on $(\Sigma , \sigma )$ that achieves the supremum

$$\begin{align*}\text{P}(\psi) := \sup_{\mu\in \mathcal{M}_\sigma}\left\{ h_\mu(\sigma) + \int \psi \ d\mu \right\}, \end{align*}$$

where $\mathcal {M}_\sigma $ is the collection of all $\sigma $ -invariant Borel probability measures on $\Sigma $ and $h_\mu (\sigma )$ denotes the (metric) entropy of $\sigma $ with respect to the measure $\mu $ [Reference Parry and Pollicott66, Theorem 3.5]. The quantity $\text {P}(\psi )$ is referred to as the pressure of $\psi $ and the measure attaining the supremum is called the equilibrium state of $\psi $ . When $\psi $ is a constant function, the measure achieving the supremum for the pressure of $\psi $ is the measure of maximal entropy. Furthermore the topological entropy $h=h(\sigma )$ of $(\Sigma ,\sigma )$ is given by $h = \text {P}(0)$ .

Consider the quantities

$$ \begin{align*} \alpha_{\min} := \inf_{\mu \in \mathcal{M}_\sigma}\int_\Sigma \psi \ d\mu \ \ \text{ and } \ \ \alpha_{\max} := \sup_{\mu \in \mathcal{M}_\sigma} \int_\Sigma \psi \ d\mu. \end{align*} $$

The large deviations principle implies (since functions that are constant on $2$ -cylinders are Hölder) that there exists a real analytic, concave function $\mathcal {L}(\psi , \cdot ) : {\mathbb {R}} \to {\mathbb {R}}_{>0}\cup \{\infty \}$ such that, for any nonempty sets $U \subset V \subset {\mathbb {R}}$ with U open and V closed we have

(2.1) $$ \begin{align} -\inf_{s\in U} \mathcal{L}(\psi, s) &\le \liminf_{n\to\infty} \frac{1}{n} \log \mu \left( x \in \Sigma : \frac{\psi^n(x)}{n} \in U \right) \nonumber \\ &\le \limsup_{n\to\infty} \frac{1}{n} \log \mu \left( x \in \Sigma : \frac{\psi^n(x)}{n} \in V \right) \le -\inf_{s\in V} \mathcal{L}(\psi,s). \end{align} $$

See [Reference Kifer56, Theorem in Section 2.1] for this result. The same result holds, with the same rate function $\mathcal {L}(\psi ,\cdot )$ , when we replace the sets

$$\begin{align*}\mu \left( x \in \Sigma : \left| \frac{\psi^n(x)}{n} - \eta \right| < \epsilon \right) \end{align*}$$

with the sequence of (normalized) cardinalities

$$\begin{align*}\frac{1}{\#\{x \in \Sigma : \sigma^n(x) = x \}}\#\left\{x \in \Sigma: \sigma^n(x) = x \text{ and } \left| \frac{\psi^n(x)}{n} - \eta \right| < \epsilon \right\}. \end{align*}$$

This is a well-known result that follows from the same proof as that of [Reference Kifer56, Theorem in Section 2.1]. The function $\mathcal {L}(\psi ,\cdot )$ is the Legendre transform of $t \mapsto \text {P}(t\psi ) - h$ . That is,

(2.2) $$ \begin{align} - \mathcal{L}(\psi,s) = \inf_{t\in {\mathbb{R}}} (\text{P}(t\psi) - h - ts ). \end{align} $$

Furthermore, $\mathcal {L}(\psi ,\cdot )$ is finite on $[\alpha _{\min }, \alpha _{\max }]$ and is infinite otherwise. An alternative characterization for $\mathcal {L}$ is the following:

$$\begin{align*}-\mathcal{L}(\psi,\eta)= \sup\left\{ h_\mu(\sigma) : \mu \in\mathcal{M}_\sigma \text{ and} \int \psi \ d\mu = \eta\right\} - h. \end{align*}$$

A function $\psi : \Sigma \rightarrow {\mathbb {R}}$ is lattice if there are $a,b \in {\mathbb {R}}$ satisfying

$$\begin{align*}\left\{ \psi^n(x) + an : x\in\Sigma \text{ and } \sigma^n(x)=x \text{ for some } n\ge 1 \right\} \subset b{\mathbb{Z}}. \end{align*}$$

If this is not the case then we say that $\psi $ is nonlattice.

2.5 Suspension flows

In this subsection we define suspension flows of subshifts of finite type. See Chapters $1$ to $6$ of [Reference Parry and Pollicott66] for more details on the results stated in this subsection. Let $\Sigma _A$ be a transitive subshift of finite type and $r: \Sigma _A \to {\mathbb {R}}_{>0}$ a function that is constant on $2$ -cylinders. We note that in [Reference Parry and Pollicott66, Chapter 6] suspension flows are considered over mixing subshifts, however the same proofs (with some minor modifications) work when the subshift is transitive. We define the suspension flow of $\Sigma _A^r$ to be the space

$$\begin{align*}\Sigma_A^r = \{(x,t) \in \Sigma_A \times {\mathbb{R}}_{\ge 0} : 0 \le t \le r(x) \} / \sim \end{align*}$$

where $(x,t) \sim (r(x),0)$ , equipped with the flow $\sigma ^r=(\sigma ^r_t)_{t>0}$ so that $\sigma _t^r$ sends $(x,s)$ to $(x,s+t)$ for $s \in {\mathbb {R}}$ . There is a natural metric on $\Sigma _A^r$ which can be constructed as in [Reference Parry and Pollicott66]. We will not present the construction of this metric here as it is a little technical.

For a Hölder continuous function $\Phi : \Sigma _A^r \to {\mathbb {R}}$ we can define its pressure as

$$\begin{align*}\text{P}_{\sigma^r}(\Phi) = \sup_{m \in \mathcal{M}_{\sigma^r}} \left\{ h_m(\sigma^r) + \int_{\Sigma_A^r} \Phi \ dm \right\}, \end{align*}$$

where $\mathcal {M}_{\sigma ^r}$ is the space of $\sigma ^r_t$ - invariant Borel probability measures on $\Sigma _A^r$ and $h_m(\sigma ^r)$ is the entropy of the time-one map $\sigma ^r_1$ for the measure m [Reference Parry and Pollicott66, Section 6].

Let $\delta _r>0$ be the unique number such that $\text {P}(-\delta _r r) =0$ and write $\mu _{-\delta _r r}$ for the equilibrium state of $-\delta _r r$ on $\Sigma _A$ . The measure of maximal entropy for $\Sigma _A^r$ is (locally) given by

$$\begin{align*}\frac{\mu_{-\delta_r r} \times \text{Leb}}{\int r \ d\mu_{-\delta_r r}}, \end{align*}$$

where $\text {Leb}$ represents the Lebesgue measure along $\mathbb {R}_{\ge 0}$ ([Reference Parry and Pollicott66, Proposition 6.1]). That is, up to normalization, the measure of maximal entropy is the measure that acts as Lebesgue along the fibers of the suspension and as $\mu _{-\delta _r r}$ on the base. If we write m for the measure of maximal entropy then we have that

$$\begin{align*}\left. \frac{d}{ds}\right|{}_{s=0} \text{P}_{\sigma^r}(s\Phi) = \int \Phi \ dm \end{align*}$$

(see [Reference Parry and Pollicott66, Proposition 4.10]).

For $T>0$ we will write $P(\Sigma _A^r, T)$ for the collection of periodic orbits of $\sigma ^r$ of length less than T. Given $R>0$ we will write $P(\Sigma _A^r, R , T)$ for the collection of periodic orbits of length between $T -R$ and $T+R$ .

Given a Hölder continuous function $\Phi : \Sigma _A^r \to {\mathbb {R}}$ it is a standard result that for any $R>0$ sufficiently large

$$\begin{align*}\lim_{T\to\infty} \frac{1}{T} \log \left(\sum_{\tau \in P(\Sigma_A^r, R, T)} e^{-s \int_\tau \Phi }\right) = \text{P}_{\sigma^r}(-s\Phi) \end{align*}$$

for any $ s\in {\mathbb {R}}$ (see for example [Reference Parry and Pollicott66, Proposition 5.10]).

Lastly we recall that two functions $\Phi , \Psi : \Sigma _A^r \to {\mathbb {R}}$ are cohomologous if $\Phi - \Psi = u'$ where $u : \Sigma _A^r \to {\mathbb {R}}$ is continuously differentiable (along flow lines) and

$$\begin{align*}u'(x) = \lim_{t\to0} \frac{u(\sigma^r_t(x)) - u(x)}{t}. \end{align*}$$

Further the function $s \mapsto \text {P}_{\sigma ^r}(s\Phi )$ is a straight line if and only if $\Phi $ is cohomologous to a constant function.

5.1 The classes $\mathfrak {G}$ and $\mathfrak {X}$

Throughout this and the next section we will work with the following notion of compatibility of pairs of group actions on $\mathrm {CAT}(0)$ cube complexes.

Note that in the definition above we do not require the action on $\mathcal {X}_\ast $ to be proper.

Since hyperplane stabilizers are always convex-cocompact and being a contracting element does not depend on the cubulation [Reference Genevois39, Lemma 4.6], it follows that $(\Gamma ,\mathcal {X},\mathcal {X}_\ast )\in \mathfrak {X}$ whenever $\Gamma \in \mathfrak {G}$ and $\mathcal {X},\mathcal {X}_\ast $ are cubulations of $\Gamma $ .

If $(\Gamma ,\mathcal {X},\mathcal {X}_\ast )\in \mathfrak {X}$ , we should think of $\mathcal {X}$ and $\mathcal {X}_\ast $ as “compatible” cubulations of $\Gamma $ with well-behaved counting properties. Compatibility is guaranteed by (2), by means of Proposition 5.7 that allows us to combine $\mathcal {X}$ and $\mathcal {X}_\ast $ into a new cubulation of $\Gamma $ . Then (1) implies that the new cubulation is also virtually co-special, allowing us to use the automaton from Example 2.5. This automaton is upgraded in Theorem 5.11 to an automaton that “remembers” both $\mathcal {X}$ and $\mathcal {X}_\ast $ . Then (3) is used to relate counting computations over this automaton to counting properties of the length functions $\ell _{\mathcal {X}},\ell _{\mathcal {X}_\ast }$ over conjugacy classes of $\Gamma $ . This counting is done in Section 6.

The main result of this subsection is the following proposition, which tells us that the class $\mathfrak {G}$ is much bigger than the class of cubulable hyperbolic groups.

For $\Gamma =W_G$ a right-angled Coxeter group, a loose square is a full subgraph $\Delta \subset G$ that is a square, and such that for every maximal subgraph $\Lambda \subset G$ with $W_\Lambda $ virtually abelian, either $\Delta \subset \Lambda $ or $\Delta \cap \Lambda $ generates a finite subgroup of $\Gamma $ .

Many nonrelatively hyperbolic right-angled Artin and Coxeter groups satisfy assumptions ii) and iii) of Proposition 5.2. Indeed, a RAAG is nontrivially relatively hyperbolic if and only if it is a nontrivial free product, and hence RAAGs with underlying graphs n-agons for $n\geq 5$ are not relatively hyperbolic and belong to $\mathfrak {G}$ .

For the proof of Proposition 5.2 we require two results, the first one being an observation about virtual specialness.

The second result we need is a criterion of convex-cocompactness for subgroups of cubulable relatively hyperbolic groups, which may be of independent interest and whose proof is postponed to the appendix.

5.2 Constructing the appropriate automaton

In this subsection we construct a finite-state automaton for a triplet $(\Gamma ,\mathcal {X},\mathcal {X}_\ast )$ in $\mathfrak {X}$ , our main result being Theorem 5.11. This is the automaton we will use to prove our main results for the length functions $\ell _{\mathcal {X}}^{\mathfrak {w}}, \ell ^{\mathfrak {w}_\ast }_{\mathcal {X}_\ast }$ in Section 6, and it implies most of Theorem 1.4. This automaton can be thought of as a cubical analog of the automaton for pairs of word metrics on hyperbolic groups constructed by Calegari-Fujiwara in [Reference Calegari and Fujiwara14, Lemma 3.8] and explained in Example 2.4.

Our starting point in the construction is the automaton for special cube complexes highlighted in Example 2.5. To use this automaton, our first step is the construction of a (virtually co-special) cubulation for $\Gamma $ that simultaneously encodes the actions on $\mathcal {X}$ and $\mathcal {X}_\ast $ . This is the content of the next proposition.

The proof of Proposition 5.7 uses the formalism of median algebras, from which refer the reader to [Reference Fioravanti34, Subsection 2.1.5]. We require the following lemma, which is a slight generalization of the implication $(3) \Rightarrow (1)$ in [Reference Fioravanti34, Proposition 7.9].