2.1 Exporting growth rates to converging sequences

We now give a proof of Theorem A.

Proof. Let $A_1,A_2$ be the subsets of the statement. By Lemma 2.5, for $i \in \{1,2\}$ , we get

\begin{align*}\frac{\omega_{A_i}(d_{S_n})}{n} \leq \omega_{A_i}(d) \leq \frac{\omega_{A_i}(d_{S_n})}{n-2-\alpha}. \end{align*}

Recall that, by assumption, we have

\begin{align*}\omega_{A_1}(d)-c \cdot \omega_{A_2}(d) \gt0. \end{align*}

On the other hand, we have, for every n,

\begin{align*}\frac{\omega_{A_1}(d_{S_n})}{n-2-\alpha}-c \cdot\frac{\omega_{A_2}(d_{S_n})}{n-2-\alpha} \geq \omega_{A_1}(d) - c \cdot\frac{\omega_{A_2}(d_{S_n})}{n} \cdot \frac{n}{n-2-\alpha} \geq \omega_{A_1}(d) - c \cdot\omega_{A_2}(d) \frac{n}{n-2-\alpha}. \end{align*}

The right-hand side is larger than 0 for large enough n. Hence, $\omega_{A_1}(d_{S_n})-c \cdot\omega_{A_2}(d_{S_n})\gt0$ .

Remark 2.6. Note that one can also obtain similar bounds in the opposite direction:

\begin{align*}\omega_{A_1}(d)-c \cdot\omega_{A_2}(d) \geq \frac{\omega_{A_1}(d_{S_n})}{n} - c \cdot\frac{\omega_{A_2}(d_{S_n})}{n-2-\alpha} = \frac{1}{n} (\omega_{A_1}(d_{S_n}) - c \cdot\frac{n}{n-2-\alpha}\omega_{A_2}(d_{S_n})). \end{align*}

We could hope to deduce from these the exponential genericity of the limit action $G \curvearrowright X$ , if we knew that the quantity on the right-hand side was positive for some n, the approximant metrics $d_{S_n}$ .

We now give the proof of Proposition 1.2.

Proof of Proposition 1.2. Since $\omega_G(d)$ is a true limit, the fact that $\omega_G(d)\gt \omega_A(d)$ gives

\begin{align*} \lim_{k \rightarrow \infty} \frac{\log \#\{g \in G: d(x_0,g x_0) \leq k\}}{k}\gt \limsup_{k \rightarrow \infty} \frac{\log \#\{g \in G: g \notin A \text{ and }d(x_0,g x_0) \leq k\}}{k}, \end{align*}

and in particular, given $\epsilon\gt0,$ there exists $n_0$ such that, for all $k \geq n_0$ , we have

\begin{align*}\frac{\log \#\{g \in G: d(x_0,g x_0) \leq k\}}{k}\gt\frac{\log \#\{g \in G: g \in A \text{ and } d(x_0,g x_0) \leq k\}}{k}+\epsilon.\end{align*}

Multiplying both sides by k and taking exponentials gives us

\begin{align*}\#\{g \in G: d(x_0,g x_0) \leq k\} \gt \#\{g \in G: g \in A \text{ and } d(x_0,g x_0) \leq k\}.e^{\epsilon.k}.\end{align*}

This yields one of the implications. The other implication follows by starting from the last inequality and going backwards (in particular, it does not use the true-limit assumption).

2.2 Exponential genericity

From the above, the proofs of Theorems B‣ and B follow readily.

Proof of Theorem B′. Let (X,d) be the geodesic metric space with strongly contracting elements on which G acts, and let A be the collection of elements which are not strongly contracting on (X,d). We have $\omega_A(d)\lt\omega_G(d)$ (by [Reference YangYan20]). Hence, using Theorem A, we obtain that $\omega_{A}(d_{S_n})\lt\omega_{G}(d_{S_n})$ for all n large enough. By Proposition 1.2, since $\omega_G(d_{S_n})$ is a true limit, we have exponential genericity.

Proof of Theorem B. Let (X,d) be the injective space on which $G={{\rm MCG}}(\Sigma)$ acts geometrically. In [Reference Sisto and ZalloumSZ24] it is shown that pseudo-Anosov mapping classes act as strongly contracting elements on these spaces (in fact, they are characterized by this property). Hence, the result follows by Theorem B′.

2.3 Improving the hierarchies

Now, we provide a proof of Theorem F.

Proof. In [Reference Petyt and ZalloumPZ24, Theorem H], Petyt and the third author construct, for each mapping class group ${{\rm MCG}}(\Sigma)$ , a geometric action on an injective space (X,d) so that each $g.x,h.x$ are connected by a hierarhcy path which is also an $\alpha$ -rough geodesic, where $\alpha$ depends only on $\Sigma$ . Now, by Lemma 2.1, we have $(n-1-\alpha)d_{S_n}(g,h)-(n-1) \leq d(gx_0,hx_0) \leq n d_{S_n}(g,h)$ for each large enough n. Dividing by n provides us with the following equation:

\begin{align*}\bigg(1-\frac{1+\alpha}{n}\bigg)d_{S_n} - \bigg(1-\frac{1}{n}\bigg) \leq \frac{d}{n} \leq d_{S_n}.\end{align*}

The metric $d_n:=({d}/{n})$ is injective for each n, and each $\alpha$ -rough geodesic in d becomes an $({\alpha}/{n})$ -rough geodesic in $d_n$ . The statement then follows easily by pushing the $({\alpha}/{n})$ -rough geodesics to $d_{S_n}$ via the above equation.

2.4 Cogrowth

Finally, we prove Theorem D.

Proof of Theorem D. Since (X,d) admits an action with a strongly contracting element, it has pure exponential growth by [Reference YangYan19]. Hence, by [Reference Choi, Gekhtman, Yang and ZhengCGY⁺24, Theorem 1.13], $\lambda_H(d) \gt \lambda_{G}(d)/2$ . Hence, by Theorem A, the result follows.

3.1 Sketch of the proof

To enhance clarity, we begin with an overall sketch of the proof.

The primary goal is to construct a generating set S with the desired property that is independent of the confined subgroup H. Given an element $(X, d) \in {\mathcal{D}\mathrm{(G)}}$ with contracting elements, we obtain $S=S_n$ by selecting a large ball of radius n in X, ensuring that the quasi-geodesic constants $(K_n, C_n)$ for the orbit map from ${\rm Cay}(G, S)$ to X satisfy that $K_n$ is sufficiently close to 1. The required closeness to 1 depends on $\omega_X(G)$ and is mediated by the constant a in Lemma 3.14 in § 3.3. Furthermore, we fix a finite set of independent elements $\mathcal{F}$ , chosen once and for all, which does not depend on the choice of S or H, but only depends on the metric d. Our goal will be to show that $\omega_{G/H}(d_{S_n}) \lt \omega_{G}(d_{S_n})$ for all n large enough.

A key tool in our proof will be the following gap criterion (§ 3.6). Roughly, for a subset $A \subset G$ , it gives conditions to construct, for each element $g\in A$ and constant $L\gt0$ , a set of $2^{\| g \|_S/L}$ many different elements in G of length approximately $\|g\|_S$ . A computation using critical exponent series then yields $\omega_A(S) \lt \omega_G(S)$ . More precisely, let $\mathcal{P}(n,L)$ denote the subsets of integer multiples of L ranging from L to $\lfloor n/L \rfloor L$ , where multiples of $L^2$ are enforced (see Definition 3.17). The gap criterion says that if, for a given $A \subset G$ , we can construct

1. (i) for every $g \in A$ , an injective map $\Phi_g \colon\mathcal{P}(|g|,L) \to G$ ;

2. (ii) so that $g_1 \neq g_2$ implies $\mathrm{Im}(\Phi_{g_1})\cap \mathrm{Im}(\Phi_{g_2}) = \emptyset$ ; and

3. (iii) so that the word length (wrt to S) of $\Phi_g(I)$ is comparable to that of g (up to a logarithmic error in L and factors depending on $|I|$ and uniform constants fixed in the beginning and depending on $\omega_G(S)$ );

then it follows that $\omega_A(S) \lt \omega_G(S)$ .

In practice, we will construct a map $\Phi_g$ satisfying condition (1) in the gap criterion, for every arbitrary element $g \in G$ , and such that $\mathrm{Im}(\Phi_g) \subset Hg$ (see § 3.4).

Specifically, $\Phi_g$ is constructed by inserting contracting elements into a word representing g (in the word metric $d_S$ ) at positions specified by $I \in \mathcal{P}(\|g\|_S, L)$ . We will call this $\Phi_g$ the extension map, since it is constructed by ‘extending’ g in multiple ways.

The injectivity of $\Phi_g$ is established in § 3.5. This requires the C-contracting axes of elements in $\mathcal{F}$ to be sufficiently long, specifically exceeding $2a \log L + D$ . Here, a is the constant alluded to above in the construction of the generating set, appearing in Lemma 3.14. Hence, it depends only on S and d and is, in particular, independent of H. On the other hand, D does depend on H.

The injectivity of the map follows from the fact that the inserted contracting axes are sufficiently long to persist after insertion. Specifically, we can detect where we inserted them by examining the fellow-traveling contracting axes of the geodesic from 1 to $\Phi_g(I)$ .

To elaborate, for each axis we insert, the fellow-travel property of the admissible path ensures that the projections of 1 and $\Phi_g(I)$ onto the axis are widely separated. Conversely, for each non-inserted axis, Lemmas 3.14 and 3.22 guarantee that the projections of 1 and $\Phi_g(I)$ onto the axis are close, giving that the length of the axis exceeds $2a\log L + D$ . Together, these two facts establish injectivity.

Finally, we will apply the gap criterion when A is a set of suitable representatives of $G/H$ (§ 3.4). In this case, the fact that $\mathrm{Im}(\Phi_g) \subset Hg$ will ensure condition (2) in the gap criterion.

The desired bound on the length of $\Phi_g(I)$ required by condition (3) follows then by the choice of a and S. Hence, the gap criterion goes through, and we obtain $\omega_{G/H}(d_S) \lt \omega_G(S)$ , as desired, finishing the proof of Theorem C′.

3.2 Preliminaries

We first introduce and recall some basic concepts and results on diameters of projections to C-contracting subsets.

3.2.1 Contracting geodesics

Let (X, d) be a geodesic metric space. Let Z be a subset of X and x be a point in X. By d(x, Z) we mean the set distance between x and Z, i.e.

\begin{align*}d(x, Z) : = \inf \{ d(x, y): y \in Z \}. \end{align*}

Let $\delta\gt0$ be a large universal constant and

\begin{align*} \pi_{Z}(x) := \{ y\in Z: d(x, y) \le d(x, Z)+\delta \} \end{align*}

be the set of closest point projections from x to Z. Then $\pi_{Z}(x)$ is non-empty. We refer to $\pi_{Z}(x) $ as the projection set of x to Z. Define $d_Z^{\pi}(x,y) := {\mathbf{diam}(\pi_Z(x) \cup \pi_Z(y))}$ for subset $Z\subset X$ and $x, y\in X$ , where ${\mathbf{diam}(\cdot)}$ denotes the diameter of a subset in the metric d.

Definition 3.1. We say a subset $Z \subseteq X$ is C-contracting for a constant $C\gt0$ if, for all pairs of points $x, y \in X$ , we have

\begin{align*}d(x, y) \leq d(x, Z) \quad \Longrightarrow \quad d_Z^{\pi}(x,y) \leq C. \end{align*}

Any such C is called a contracting constant for Z. A collection of C-contracting subsets shall be referred to as a C-contracting system. We note that $d^{\pi}_Y$ satisfies the triangular inequality

\begin{align*}d^{\pi}_Y(x, z) \le d^{\pi}_Y(x, y) + d^{\pi}_Y(y, z)\end{align*}

for any $x, y, z\in X$ .

An element $h\in \mathrm{Isom}(X)$ is called contracting if it acts by translation on a contracting bi-infinite quasi-geodesic. Equivalently, the map $n\in \mathbb Z\longmapsto h^no$ is a quasi-geodesic with a contracting image.

Given a contracting element h, there exists a maximal elementary group E(h) containing $\langle h \rangle$ as a finite-index subgroup. Precisely,

\begin{align*}E(h) := \{ g \in G : \exists n \gt0, (gh^ng^{-1}=h^n) \mbox{ or } (gh^ng^{-1}=h^{-n}) \}. \end{align*}

We will define

\begin{align*}\mathrm{Ax}(h) := \{ f \cdot o | f \in E(h) \} \end{align*}

and denote it the axis of h.

Definition 3.2. Let $\gamma: [0, 1]\to X$ be a path in X. Then $\gamma$ is called (K, C)-quasi-geodesic if, for any $0\le a \lt b \le 1$ , $length(\gamma|_{[a,b]})\le K d(\gamma(a), \gamma(b)) + C$ , where $\gamma|_{[a,b]}$ is the subpath of $\gamma$ with endpoints $\gamma(a)$ and $\gamma(b)$ .

Let G be a group with a finite generating set S and Cayley graph $\Gamma$ . Let $[g, h]_S$ be the geodesic in $\Gamma$ with endpoints g and h. Denote the word metric on $\Gamma$ by $d_S$ .

Fix a basepoint $o\in X$ . Let $\pi: G\to X$ be defined by $g\mapsto go$ . Then $\pi([g, h]_S)$ is a (K, C)-quasi-geodesic for some $K\ge 1$ , $C\ge 0$ . Contracting subsets are Morse, and are preserved up to taking finite Hausdorff distance. The following properties shall be used later on.

Lemma 3.3. Let $Y\subseteq X$ be a C-contracting subset for $C\gt0$ . Then the following holds.

1. (i) For any geodesic $\gamma$ , we have

   \begin{align*} |d^{\pi}_Y(\gamma^-,\gamma^+)- {\mathbf{diam}(\gamma\cap N_C(Y))}|\leq 4C.\end{align*}

2. (ii) There exists $\hat C=\hat C(C)$ such that $d^{\pi}_Y(y, z)\le d(y, z)+\hat C$ for any $y,z\in X$ .

3. (iii) There exists $\hat C=\hat C(C)$ such that for any geodesic $\gamma$ , we have

   \begin{align*}|{\mathbf{diam}(\pi_Y(\gamma))} - d^{\pi}_Y(\gamma_-, \gamma_+) | \leq \hat C.\end{align*}

We refer the reader to [Reference YangYan22] for a proof.

A C-contracting system $\mathcal{X}$ has the bounded intersection property if there exists a constant $B \geq 0$ so that ${\mathbf{diam}(\pi)}_{X}(X') \leq B$ for every $X,X' \in \mathcal{X}$ with $X' \neq X$ . For example, it is easy to see that the collection $\{ g \mathrm{Ax}(h) : g \in G\}$ is a contracting system with bounded intersection.

Definition 3.4. We say that a collection of contracting elements $\mathcal{F} = \{ f_i \}_{i \in \mathcal{I}} \subset G$ are independent if, for every $i, j \in \mathcal{I}$ with $i \neq j$ , the collection

\begin{align*}\{ g\mathrm{Ax}(f_k) : g \in G, k\in \{i,j \}\} \end{align*}

is a contracting system with bounded intersection.

3.2.2 Extension Lemma

In this article, we frequently construct a path labeled by a word $(g_1, g_2,\ldots,g_n)$ , which, by convention, means the following concatenation:

\begin{align*}[o,g_1o]\cdot g_1[o,g_2o]\cdots (g_1\ldots g_{n-1})[o,g_no],\end{align*}

where the basepoint o is understood in context. With this convention, the paths labeled by $(g_1,g_2,g_3)$ and $(g_1g_2, g_3)$ , respectively, differ depending on whether $[o,g_1o]g_1[o,g_2o]$ is a geodesic or not.

We fix a finite set $F \subseteq \Gamma$ of independent contracting elements and let $\mathscr F = \{g \mathrm{Ax}(f) : f \in F, g \in \Gamma\}$ . The following notion of an admissible path allows us to construct a quasi-geodesic by concatenating geodesics via $\mathscr F$ .

Definition 3.5 (Admissible path). Given $L,\tau\geq0$ , a path $\gamma$ is called $(L,\tau)$ -admissible in X if $\gamma$ is a concatenation of geodesics $p_0q_1p_1\cdots q_np_n$ $(n\in\mathbb{N})$ , where the two endpoints of each $p_i$ lie in some $X_i\in \mathscr F$ , and the following long local and bounded projection properties hold.

(LL) Each $p_i$ for $1\le i\lt n$ has length bigger than L, and $p_0,p_n$ could be trivial.

(BP) For each $X_i$ , we have $X_i\ne X_{i+1}$ and $\max\{{\mathbf{diam}(\pi_{X_i}(q_i))},{\mathbf{diam}(\pi_{X_i}(q_{i+1}))}\}\leq\tau$ , where $q_0:=\gamma^-$ and $q_{n+1}:=\gamma^+$ by convention.

The collection of sets $\{X_i: 1\le i\le n\}$ will be referred to as the contracting subsets associated to the admissible path.

Remark 3.6. The path $q_i$ could be allowed to be trivial, so by the (BP) condition, it suffices to check $X_i\ne X_{i+1}$ . It will be useful to note that admissible paths could be concatenated as follows. Let $p_0q_1p_1\cdots q_np_n$ and $p_0'q_1'p_1'\cdots q_n'p_n'$ be $(L,\tau)$ -admissible. If $p_n=p_0'$ has length bigger than L, then the concatenation $(p_0q_1p_1\cdots q_np_n)\cdot (q_1'p_1'\cdots q_n'p_n')$ has a natural $(L,\tau)$ -admissible structure.

We frequently use the following lemma to show $X_i\neq X_{i+1}$ in property (BP).

Lemma 3.7. Using notations in the Definition 3.5 of the admissible path, for all $1\le i\le n$ , if

$\max\{ {\mathbf{diam}(\pi_{X_{i}}(q_{i}))}, {\mathbf{diam}(\pi_{X_{i-1}}(q_{i}))}\}\leq\tau$ and $\ell(q_{i}) \gt \tau$ , then $X_i \neq X_{i-1}$ .

Proof. Suppose to the contrary that $X_i = X_{i-1}$ . Let $a = (q_i)^-$ and $b = (q_i)^+$ . Then $a = (p_{i-1})^+$ , $b = (p_i)^-$ . Since $p_i\subset X_i$ , $p_{i-1}\subset X_{i-1}$ , we have $a, b\in X_i$ . But $d(a, b) = d(\Pi_{X_i}(a), \Pi_{X_i}(b)) \lt {\mathbf{diam}(\Pi_{X_i}(q_i))} \le \tau$ . Since a,b are the endpoints of $q_i$ , $\ell(q_i)\le \tau$ . This contradicts our assumption that $\ell(q_i) \gt \tau$ .

A sequence of points $x_i$ in a path p is called linearly ordered if $x_{i+1}\in [x_i, p^+]_p$ for each i.

Definition 3.8 (Fellow travel). Let $\gamma = p_0 q_1 p_1 \cdots q_n p_n$ be an $(L, \tau)-$ admissible path. We say $\gamma$ has the r-fellow-travel property for some $r\gt0$ if, for any geodesic $\alpha$ with the same endpoints as $\gamma$ , there exists a sequence of linearly ordered points $z_i,w_i$ ( $0 \le i \le n$ ) on $\alpha$ such that

\begin{align*}d(z_i, p_{i}^-) \le r,\quad d(w_i, p_{i}^+) \le r.\end{align*}

In particular, ${\mathbf{diam}(N_r(X_i)\cap \alpha)} \ge L$ for each $X_i\in \mathscr F(\gamma)$ .

The following result says that a local long admissible path enjoys the fellow-travel property.

Proposition 3.9 [Reference YangYan14]. For any $\tau\gt0$ , there exist $L, r, c\gt0$ depending only on $\tau,C$ such that any $(L, \tau)$ -admissible path has the r-fellow-travel property. In particular, it is a c-quasi-geodesic.

The next lemma gives a way to build admissible paths.

Lemma 3.10 (Extension lemma).

For any independent contracting elements $h_{1}, h_{2}, h_{3} \in G$ , there exist constants $L, r, B\gt0$ depending only on C with the following property.

Choose any element $f_i\in \langle h_i\rangle$ for each $1\le i\le 3$ to form the set F of $f_i$ satisfying $\min \{\| f_i \|_X \} \geq L$ . Let $g,h\in G$ be any two elements.

There exists an element $f \in F$ such that the path

\begin{align*}\gamma:=[o, go]\cdot(g[o, fo])\cdot(gf[o,ho])\end{align*}

is an $(L, \tau)$ -admissible path relative to $\mathscr F$ .

Remark 3.11. Since admissible paths are local conditions, we can connect via F any number of elements $g\in G$ to satisfy (1) and (2). We refer the reader to [Reference YangYan14] for a precise formulation.

Corollary 3.12 [Reference YangYan14, Corollary 3.9]. Let $\gamma$ be an admissible path and $X_k$ be a contracting subset for $\gamma$ , where $0\le k\le n$ . Then there exists $N = N(C, \tau) \gt 0$ such that ${\mathbf{diam}(\Pi_{X_k}([\gamma_{-}, (p_k)_{-}]_{\gamma}))} \lt N$ , where $\gamma_{-}$ and $(p_k)_{-}$ denote the starting endpoints of $\gamma$ and $p_k$ , respectively.

The following lemma allows us to find a contracting element that is independent of the non-degenerate confining subset.

Lemma 3.13 [Reference Ding and YangDY24]. Let P be a finite non-degenerate set. Then there exists a finite set F of independent contracting elements in G so that, for any $p\in P$ and $f\in F$ , we have $p \mathrm{Ax}(f)\ne \mathrm{Ax}(f)$ .

3.3 Projection of quasi-geodesics to contracting subsets

The results in this section will allow us to bound the diameter of the projection of quasi-geodesic to contracting subsets of Y. The first result shows the bound on the diameter can be improved arbitrarily as the multiplicative constant K in the quasi-isometry goes to 1.

Lemma 3.14. For any $\beta\gt0$ , $C\ge 0$ , and $K\ge 1$ , there exist $a = a(K, \beta) \ge 0$ , $b = b(\beta, K, C) \gt 0$ , and $c = c(K) \gt 0$ with the following property. Let Y be a $\beta$ -contracting subset of X and $\gamma$ be a (K, C)-quasi-geodesic with endpoints $x\in X$ and $y\in Y$ . Then

\begin{align*}{\mathbf{diam}(\pi_{Y}(\gamma))} \le a \log d(x, y) + b + c d(\pi_Y(x), y)).\end{align*}

Furthermore, we may choose $a(1, \beta)=0$ and $\lim_{K\to 1} a(K, \beta) = 0$ for any $\beta \gt 0$ .

Proof of the claim. Let $D = {\mathbf{diam}(\pi_Y(\gamma_1))}$ , $l = \mathrm{length}(\gamma_1)$ , and $h = d(x, Y)$ . Then, by the triangular inequality,

\begin{align*}d(x, z) \le d(x, Y) + d(z, Y) + d^{\pi}_Y(x, z) \le h +\theta h + D.\end{align*}

Since $\gamma$ is a (K, C)-quasi-geodesic, we have

(3.1) \begin{align} l \le K d(x, z) + C \le K(h(1+\theta) + D) + C. \end{align}

We are going to bound D. To that end, we write $\gamma_1 = \gamma_2 \cup \gamma_3$ as the concatenation of two paths where the subpath $\gamma_2$ has length h starting at x and thus is contained in B(x, h), and $\gamma_3$ is the remaining subpath of $\gamma_1$ with $\mathrm{length}(\gamma_3) = l - h$ . Since Y is $\beta$ -contracting and $h = d(x, Y)$ , we have ${\mathbf{diam}(\pi_Y(\gamma_2))} \le \beta$ . For $\gamma_3$ , we may divide it into disjoint subpaths of length $\theta h$ , and the number of them is at most $({l-h})/{\theta h} + 1.$ Since the path $\gamma_1$ from x to z has distance at least $\theta h$ to Y, we have that each of these subpaths is contained in a ball which is disjoint from Y. Thus,

\begin{align*}{\mathbf{diam}(\pi_Y(\gamma_3))}\le \beta({(l-h)}/{\theta h} + 1).\end{align*}

Taking into account the projections of $\gamma_2$ and $\gamma_3$ , we have

\begin{align*}D = {\mathbf{diam}(\pi_Y(\gamma_1))} \le {\mathbf{diam}(\pi_Y(\gamma_2))} + {\mathbf{diam}(\pi_Y(\gamma_3))} \le \beta (2 + {(l-h)}/{\theta h}),\end{align*}

which yields

\begin{align*}l \ge h + ({D}/{\beta} - 2)\theta h.\end{align*}

With (3.1), we have

\begin{align*}h +({D}/{\beta} - 2) \theta h \le K(h(1+\theta) + D) + C,\end{align*}

or equivalently, using $\theta=1-({1}/{K})$ ,

\begin{align*}D({\theta h}/{\beta} - K) \le h(K +K\theta- 1+2\theta ) + C = 2h(K +1)\theta + C.\end{align*}

Recall that $\mathfrak h=\max\{({2K\beta})/{\theta}, {C}/({(K+1)\theta}), 1\}$ . If $h \gt \mathfrak h$ , we have ${h\theta }/{\beta} \ge 2K$ and $C\le h (K +1) \theta$ . Thus,

\begin{align*}D \le \frac{3h(K +1)\theta}{{\theta h}/{2\beta}} = 6(K + 1)\beta.\end{align*}

This proves the claim.

Let $x_0 = x$ and define inductively $x_k$ to be the first point on $\gamma$ with $d(x_k, Y) \le \theta d(x_{k-1}, Y)$ for $k\ge 1$ . Let m be the minimal k such that $d(x_k, Y)\le \mathfrak h$ . Then

\begin{align*}-{\log_{\theta} \frac{d(x,Y)}{\mathfrak h}} \le m \le 1- {\log_\theta \frac{d(x,Y)}{\mathfrak h}} \le 1 - \log_\theta d(x,Y).\end{align*}

Let $\gamma_k$ be the subpath of $\gamma$ between $x_{k-1}$ and $x_k$ . As $d(\gamma_{k}, Y)\ge \mathfrak h$ for $k\le m$ , by the claim above, we have ${\mathbf{diam}(\pi_Y(\gamma_k))}\le 6(K+1) \beta$ . If $\alpha_1$ denotes the subpath of $\gamma$ between x and $x_m$ , then

\begin{align*}{\mathbf{diam}(\pi_Y(\alpha_1))} \le 6m(K+1) C,\end{align*}

and especially $d_Y^\pi(x, x_m) \le 6m(K+1) \beta$ . Thus, $d(x_m, y)\le d(x_m, \pi_Y(x_m)) + d_Y^\pi(x, x_m) + d(\pi_Y(x), y) \le 6m(K+1)\beta + \mathfrak h + d(\pi_Y(x), y)$ . Let $\alpha_2$ be the subpath of $\gamma$ between $x_m$ and y. Since $\alpha_2$ is also a (K, C)-quasi-geodesic, we have $\mathrm{length}(\alpha_2)\le K d(x_m, y) + C$ . So, by Lemma 3.3,

\begin{align*}{\mathbf{diam}(\pi_Y(\alpha_2))} \le \mathrm{length}(\alpha_2) + \hat C \le 6m K(K+1) \beta + K \mathfrak h + K d(\pi_Y(x), y) + C + \hat C.\end{align*}

Finally,

\begin{align*}\begin{aligned} {\mathbf{diam}(\pi_Y(\gamma))} &\le {\mathbf{diam}(\pi_Y(\alpha_1))} + {\mathbf{diam}(\pi_Y(\alpha_2))}\\ &\le 6m (K+1)^2 \beta + K \mathfrak h + K d(\pi_Y(x), y) + C + \hat C\\ &\le a \log d(x, y) +b + c d(\pi_Y(x), y), \end{aligned}\end{align*}

where $a = -{6(K+1)^2 \beta}/{\log \theta}$ , $b = {6(K+1)^2 \beta} + K \mathfrak h + C + \hat C$ , and $c = K$ .

Noticing $\lim_{K\to 1^+} a = 0$ , the lemma thus follows.

3.4 Construction of extension map

In this section we will construct, for any $g \in G$ , the ‘extension map’ $\Phi_g \colon \mathcal{P}(n, L) \to Hg$ required in the ‘gap criterion’. This section follows [Reference Ding and YangDY24] closely. The main difference lies in that here, the choice of the contracting axis depends on the local structure of I.

This lemma follows directly from [Reference YangYan19, Lemmas 2.12 and 2.17].

In Lemma 3.16, we pick $n=5$ and get a finite set $\mathcal{F}$ consisting of contracting element f, and the choice of $\mathcal{F}$ is independent of the generating set S and of the normal subgroup H. This set will be fixed once and for all.

Now, let P be a finite confining subset for the confined subgroup H. Assume that P is non-degenerate, i.e. disjoint with E(G). By Lemma 3.13, we pick a contracting element $f\in G$ such that ${\mathbf{diam}(\pi_{\mathrm{Ax}(f)}([o, po]))} \le B$ for any $p\in P$ . Here, f depends on P.

Recall that, by Proposition 3.9, there exist $\mathcal{D}\gt0, \lambda\gt1, r\gt0$ such that any $(\mathcal{D}, B)$ -admissible path of $\max\{C, C'\}$ -contracting subsets has the r-fellow-travel property and is an $(\lambda, \lambda)$ -quasi-geodesic. By taking a high power, we may assume that $\|f\|_X \gt \mathcal{D}$ and $\|f_i\|_X \gt \mathcal{D}$ for each $f_i\in \mathcal{F}$ .

The following natural definition will be important in order to get enough control when approximating X by generating sets.

Let $g\in G$ be an arbitrary element of G. If $\|g\|_S=n$ , $s_i \in S$ for $i=1,\ldots,n$ , and we have $g=s_1\cdots s_n$ which is a word geodesic of g. Denote $g_0 = \rm{id}$ and $g_i = s_1 \cdots s_i$ for $1\le i\le n$ .

This construction modifies the extension map in [Reference Ding and YangDY24]. Roughly, for each element $g \in G$ , this map $\Phi_g$ yields $2^{\lfloor n/L \rfloor}$ many elements in the coset of g, each with length at most $n(1 + ({1}/{L}))$ . It does so by ‘extending’ g inserting words from axes of contracting elements at prescribed places (see Figure 1 for an illustration).

Figure 1.

The construction of $\Phi_g(I)$ , where $h = (g_{a_i} f_i f) p_i (g_{a_i} f_i f)^{-1}\in H$ .

Definition 3.18. Let $g \in G$ , $g= s_1 \cdots s_n\in G$ , be a word geodesic and $L\ge 1$ . The extension map $\Phi_g: \mathcal P(n,L)\to G$ is defined by

\begin{align*}I=(a_1, \ldots, a_m)\mapsto \bigg(\prod_{i=1}^m g_{a_{i-1}}^{-1} g_{a_i} f_i f p_i f^{-1} f_i^{-1}\bigg) g_{a_{m}}^{-1} g_n,\end{align*}

where $f_i\in \mathcal F$ are chosen with the following properties:

1. (i) $[g_{a_{i-1}} o, g_{a_i} o]$ has B-bounded projection to $g_{a_i}\mathrm{Ax}(f_i)$ ;

2. (ii) $[g_{a_{i}} o, g_{a_{i+1}} o]$ has B-bounded projection to $g_{a_i}\mathrm{Ax}(f_i)$ ;

3. (iii) $[g_{a_{i} - L^2} o, g_{a_{i}} o]$ has B-bounded projection to $g_{a_i}\mathrm{Ax}(f_i)$ ;

4. (iv) $[g_{a_{i}} o, g_{L^2 + a_{i}} o]$ has B-bounded projection to $g_{a_i}\mathrm{Ax}(f_i)$ ;

5. (v) [o, f o] has B-bounded projection to $\mathrm{Ax}(f_i)$ .

   Here, $a_i = 0$ for $i\le 0$ , $a_{j} = n$ for $j\ge m+1$ by convention, and

6. (vi) $p_i\in P$ are chosen in the confining subset P so that $g_{a_i}f_i f p_i(g_{a_i}f_if)^{-1}$ is contained in H.

Let $\gamma_g(I)$ be the path labeled by

\begin{align*}(g_{a_1}, f_1,1, f, p_1, f^{-1}, 1, f_1^{-1}, \ldots, g_{a_{i-1}}^{-1} g_{a_i}, f_i, 1, f, p_i, f^{-1}, 1, f_i^{-1}, \ldots, g_{a_{m}}^{-1} g).\end{align*}

Remark 3.19. The existence of $f_i$ relies on the choice of $\mathcal{F}$ , where we take the 5-tuple to be

\begin{align*}(g_{a_i}^{-1}g_{a_{i-1}}, g_{a_i}^{-1}g_{a_{i+1}}, g_{a_i}^{-1}g_{a_i-L^2}, g_{a_i}^{-1}g_{a_i+L^2}, f).\end{align*}

The containment of the image of $\Phi_g$ in Hg crucially uses property (6) of the confined subgroup H; cf. [Reference Ding and YangDY24, Lemma 4.6].

If we identify, using the indicator function $\chi_I$ , the subset I of $\{1,\ldots,n\}$ as a sequence $(\epsilon_i)_{i=1}^n$ in $\{0,1\}^n$ by $\epsilon_i = \chi_I(i)$ , then we can write compactly $\Phi_g(I) = \prod_{i=1}^{n} s_i (g_i q_i g_i^{-1})^{\epsilon_i}$ , where $g_{a_i} = f_i f$ , $q_{a_i} = p_i$ for $1\le i\le m$ , and $g_j = q_j = id$ if $j\neq a_i$ for any $1\le i\le m$ .

Now assume that $\Phi_g(I)\in Hg$ whenever $|I|=m-1$ . Let $I\in\mathcal{P}(n, L)$ with $|I|=m$ , and write $I = \{a_1,\ldots, a_m\}$ , where $1\le a_1\lt a_2\lt\cdots\lt a_m\le n$ . Let $J = I\setminus\{a_1\} = \{a_2, \ldots, a_m\}$ . Then $|J|=m-1$ and we have

\begin{align*}\Phi_g(I) = g_{a_1} f_1 f p_1 f^{-1} f_i^{-1}g_{a_1}^{-1}\Phi_g(J) = (g_{a_1}f_1 f) p_1 (g_{a_1}f_1 f)^{-1} \Phi_g(J).\end{align*}

By the induction hypothesis, $\Phi_g(J)\in Hg$ . Furthermore, property (6) of Definition 3.18 implies $(g_{a_1}f_1 f) p_1 (g_{a_1}f_1 f)^{-1}\in H$ . Thus, $\Phi_g(I)\in Hg$ . This completes the induction step, and thus the lemma holds.

By the construction of $\gamma_g(I)$ , it suffices to show that $\mathrm{Ax}(f_i) \neq g_{a_{i}}^{-1} g_{a_{i+1}} \mathrm{Ax}(f_{i+1})$ .

If not, let $h = g_{a_{i}}^{-1} g_{a_{i+1}}$ and $Y = \mathrm{Ax}(f_i) = h \mathrm{Ax}(f_{i+1})$ . Then $o, po\in Y$ . So $\|h\|_X = d(o, h o) \le d^{\pi}_Y([o, po]) \le B$ by condition (ii) in Definition 3.18. However, since $L\gt L_1$ and $\|h\|_S \ge L$ , we have $\|h\|_X \gt B$ , which yields a contradiction.

3.5 Injectivity of extension map

In this subsection, we will prove that the extension map $\Phi_g$ introduced before is injective in $\mathcal{P}(n, L)$ . Properties (1)–(5) in Definition 3.18 will all be used to establish this, as we will see.

First, we need a preliminary result on the bounds of the projections onto two contracting subsets. This result implies that for any point, either one of the two projections lies near the common perpendicular of the two contracting sets, provided that the sets are far away from each other.

Lemma 3.22. Let Y, Z be C-contracting subsets with B-bounded projection. Let $\tau \gt 0$ . Let $y\in Y$ and $z\in Z$ such that $d^\pi_Y(y, z)\le \tau$ , $d^\pi_Z(y, z)\le \tau$ , and $d(y, z) \gt 2\tau + 10C + B$ . Then there exists $E = E(C, B)$ such that

\begin{align*}\min\{d^\pi_Y(x, y), d^{\pi}_Z(x, z)\} \le \tau + E\end{align*}

for any $x\in X$ .

Proof. Let $E = 2C + \hat C + B \gt C$ . Suppose the opposite holds, that is, $d^\pi_Y(x, y) \gt \tau + E$ , $d^{\pi}_Z(x, z) \gt \tau + E$ . Then $d^{\pi}_Z(x, y) \ge d^\pi_Z(x, z)- d^\pi_Z(y, z) \gt E \gt C$ , which means that $[x, y]\cap N_C(Z)\neq\emptyset$ . Noticing that $d(y, Z) \ge d(y, z) - d^\pi_Z(y, z) \gt E \gt C$ , let p be the first point on [y, x] such that $d(p, Z)\le C$ . Let $q \in \pi_Z(p)$ and $y' \in \pi_Z(y)$ . Then, by the C-contracting property of Z, we have $d(q, y') = d^\pi_Z(y, p) \le C$ . Since Y, Z has B-bounded projection, $d^\pi_Y(q, z) \le B$ . Considering $d(z, Y) \ge d(z, y) - d^{\pi}_Y(z, y) \ge \tau + 10C + B$ , by the triangular inequality we have $d(q, z) \le d(q, y') + d^\pi_Z(y, z) \le \tau + C$ , $d(q,Y) \geq d(z,Y) - d(q,z) - d^\pi_Y(q,z)$ , and hence

\begin{align*}d(q, Y) \ge (\tau + 10C + B) - (\tau + C) - B = 9C.\end{align*}

Thus,

\begin{align*}d(p, Y) \ge d(q, Y) - d(p, q) \ge 8C.\end{align*}

We claim that $[p, x]\cap N_C(Y) = \emptyset$ .

To prove the claim, assume, for the sake of contradiction, that v is the first point on [p,x] such that $d(v, Y) \le C$ . Noticing that $[y, p]\cap N_C(Y)\neq \emptyset$ , let u be the last point on [y, p] such that $d(u, Y) \le C$ . Then, by the C-contracting property of Y, $d^\pi_Y(u, v)\le C$ and $d^\pi_Y(v, p)\le C$ . So, by the triangular inequality, we have

\begin{align*}d(u, v) \le d(u, Y) + d(v, Y) + d^\pi_Y(u, v) \le 3C.\end{align*}

Now, since $y\in [u, v]$ , we have

\begin{align*}d(u, v) \ge d(v, p) \ge d(p, Y) - d(v, Y) - d^\pi_Y(v, p) \ge 8C - C - C = 6C \gt 3C,\end{align*}

which is a contradiction.

By the claim, we have $d^\pi_Y(p, x) \le C$ . So, by the triangular inequality, we have

\begin{align*}d^\pi_Y(y, p)\ge d^\pi_Y(y,x) - d^\pi_Y(x,p) \gt \tau + E - C.\end{align*}

Then, by Lemma 3.3, we get $d^\pi_Y(q, y) \ge d^\pi_Y(p, y) - d(p, q) - \hat{C} \ge \tau + E - 2C - \hat C$ . Thus, by the triangular inequality, it follows that

\begin{align*}d^\pi_Y(q, z)\ge d^\pi_Y(q, y)-d^\pi_Y(y,z) \gt E - 2C - \hat C.\end{align*}

However, recall that $q,z\in Z$ , so $d^\pi_Y(q, z)\le B$ follows from the B-bounded projection property. Since we took $E - 2C - \hat C = B$ , we get a contradiction, and the lemma follows.

Then we will prove the injectivity of $\Phi_g$ .

Let $g = s_1 \cdots s_n\in G$ be a word geodesic. For any $L\gt L_2$ , if $\|f\|_X \ge 2a\log (L) + D$ for each $f\in \mathcal{F}$ , then $\Phi_g: \mathcal{P}(n,L)\to Hg$ is injective.

Proof. Let b, c be the constant in Lemma 3.14 and let E be in Lemma 3.22. Let $L_2\gt0$ be large enough such that

\begin{align*}LM^{-1} - M - 2(a\log (2ML^2) + b + cB) \gt 10 C + B\end{align*}

holds for each $L \gt L_2$ , and D be a large constant to be defined below.

Let $I_1, I_2 \in \mathcal{P}(n, L)$ with $I_1\ne I_2\in \mathcal{P}(n,L)$ and so that $h := \Phi_g(I_1) = \Phi_g(I_2)$ . Let $I_1 = \{a_1, \ldots, a_{\alpha}\}$ and $I_2 = \{b_1, \ldots, b_{\beta}\}$ . Let $f_i$ ( $1\le i\le \alpha$ ) be the chosen contracting elements for $a_i\in I_1$ and $f_j'$ ( $1\le j\le \beta$ ) be the chosen contracting elements for $b_j\in I_2$ . We shall prove inductively that for each $1\le k\le \lfloor{n}/{L}\rfloor$ ,

\begin{align*}I_1 \cap \{L, 2L, \ldots, kL\} = I_2 \cap \{L, 2L, \ldots, kL\}\end{align*}

and simultaneously, $f_i = f_i'$ for each $1\le i\le \max\{\alpha,\beta\}$ . This clearly implies $I_1=I_2$ and $\alpha = \beta$ , so the proof of injectivity follows.

We start by proving the base step $k=1$ . The following claim is the main observation in the base step and the subsequent induction.

Proof of the claim. If the claim is false, without loss of generality, we assume $L\in I_1$ and $L\not\in I_2$ . Then $a_1 = L$ and $2L\le b_1 \le L^2$ since $L^2\in I_2$ . To derive a contradiction, the idea is to first give a lower bound on $d^{\pi}_Y(o, h o)$ using the admissible path $\gamma_g(I_1)$ , and then an upper bound using $\gamma_g(I_2),$ where Y is a C-contracting subset of X. Recall that $g_i = s_1 s_2\cdots s_i$ for each $1\le i\le n$ and $g_0 = \rm{id}$ . Denote by $Y = g_L \mathrm{Ax}(f_1)$ the C-contracting subset associated with $[g_L o, g_L f_1 o]$ in the admissible path $\gamma_g(I_1)$ . By the r-fellow-travel property, the two endpoints of $[g_L o, g_L f_1 o]$ are contained in the r-neighborhood of [o, ho], so there exist $u, v\in [o, ho]$ such that $d(u, g_L o)\le r$ , $d(v, g_L f_1 o)\le r$ . If $u' = \pi_Y(u)$ and $v' = \pi_Y(v)$ , the triangular inequality shows

\begin{align*}d(u', g_L o)\le d(u', u) + d(u, g_L o) = d(u, Y) + d(u, g_L o)\le 2 d(u, g_L o) \le 2r.\end{align*}

Similarly, $d(v', g_L f_1 o)\le 2r$ . Thus, we obtain

\begin{align*}d^{\pi}_Y(u, v) \ge d(u', v') \ge d(g_L o, g_L f_1 o) - d(u', g_L o) - d(v', g_L f_1 o) \ge \|f_1\|_X - 4r,\end{align*}

which yields, by Lemma 3.3,

(3.2) \begin{equation} d^{\pi}_Y(o, h o) \ge {\mathbf{diam}(\pi_Y([o, ho]))} - \hat C \ge d^{\pi}_Y(u, v) - \hat C \ge \|f_1\|_X - 4r - \hat C. \end{equation}

For a contradiction, we use the other path $\gamma_g(I_2)$ to derive an upper bound for $d^{\pi}_Y(o, h o)$ . Let $t:=b_1\le L^2 $ be the minimal integer in $I_2$ . Our idea is to give a lower bound on $d(g_t o, Y)$ then use Lemma 3.22 to give an upper bound on the projection. By construction of $f_1$ as in Definition 3.18(iv), we have

\begin{align*}d^{\pi}_Y(g_L o, g_{L^2+L} o) \le B.\end{align*}

Since $\pi$ is an (M, M)-quasi-isometry, we have $M^{-1}(j-i) - M\le d(g_i o, g_j o)\le M(j-i) + M$ . According to Lemma 3.14, since $\pi([g_L, g_{L+L^2}]_S)$ is a (K, C)-quasi-geodesic containing $g_{b_1}o=g_to$ and $d(g_L o, g_{L+L^2} o)\le M L^2 + M$ , we have

\begin{align*}d^{\pi}_Y (g_L o, g_{t} o) \le a\log(ML^2 + M) + b + cB \le a\log(2ML^2) + b + cB.\end{align*}

On the other hand, denote by $Z = g_t\mathrm{Ax}(f_1')$ the C-contracting subset associated with $[g_t o, g_t f_1' o]$ in the admissible path $\gamma_g(I_2)$ . Then, by the same reasoning as above, noticing that by Definition 3.18(iii), $d^\pi_Z (o, g_t o)\le B$ , we have

\begin{align*}d^{\pi}_Z (g_L o, g_{t} o) \le a\log(2ML^2) + b + cB.\end{align*}

From here, we are going to apply Lemma 3.22. Let $\tau = a\log(2ML^2) + b + cB$ , $y = g_L o$ , $z = g_t o$ . Then, by the equalities above, we have $d^\pi_Y(y, z)\le \tau$ and $d^\pi_Z(y, z)\le \tau$ . By choice of $L_2$ and $L \gt L_2$ , we have $d(y, z) \ge LM^{-1} - M \gt 2\tau + 10C + B$ . Clearly, $Y\neq Z$ for $d^\pi(y, z) \lt d(y, z)$ and thus Y, Z have B-bounded projection. Since $\gamma_g(I_2)$ has the r-fellow-travel property, we have $d^\pi_Z(z, h o) \ge \|f_1'\| - r \gt \tau + E$ . Suppose we have $D \gt \tau - 2a\log(L) + E + r = a\log(2M) + b + cB + E + r$ . Then, by Lemma 3.22, $d^{\pi}_Y(h o, g_L o) \le \tau + E$ .

Also, property (i) in Definition 3.18 implies $d^{\pi}_Y (o, g_L o)\le B$ . Hence, the triangular inequality for $d^{\pi}_Y$ implies

(3.3) \begin{equation} d^{\pi}_Y(o, h o)\le d^{\pi}_Y(o, g_L o) + d^{\pi}_Y(h o, g_L o)\le \tau + E + B. \end{equation}

Finally, combining the inequalities (3.2) and (3.3), we have

\begin{align*}\|f_1\|_X - 4r - \hat C \le d^{\pi}_Y(o, h o) \le \tau + E + B.\end{align*}

If we set $D = 4r + \hat C + a \log (2 M) + b + cB + E + B \gt \tau - 2a\log(L) + E + r$ , this contradicts our assumption that $\|f_1\|_X \gt 2a\log L + D$ . So the claim holds.

If $1\in I_1\cap I_2$ , it remains to prove $f_1=f_1'$ for the base case. Indeed, if $f_1\neq f_1'$ , then by the choice of $\mathcal{F}$ , $g_L \mathrm{Ax}(f_1)$ and $g_L \mathrm{Ax}(f_1')$ have the bounded projection property. For a path $\gamma$ , denote the subpath of $\gamma$ with endpoints $x, y\in \gamma$ by $\gamma|_{[x, y]}$ . Let $\gamma' = \gamma_g(P_2)|_{[h o, g_L o]} \cup \gamma_g(P_1)|_{[g_L o, h o]}$ . Then $\gamma'$ is a (D, B)-admissible path. Thus, it is a $\lambda$ -quasi-geodesic loop at h o, so $\mathrm{length}(\gamma') \le \lambda^2$ . This contradicts $\mathrm{length}(\gamma') \ge \mathrm{length}(f_1) \gt F \gt \lambda^2$ . So $f_1 = f_1'$ . This finishes the proof of the base case $k = 1$ .

Let us consider $k \ge 2$ . By the induction hypothesis, $I:= I_1 \cap \{L, \ldots, (k-1)L\} = I_2 \cap \{L, \ldots, (k-1)L\}$ and $f_i = f_i'$ for $a_i\le k-1$ . Let $x = g_{(k-1)L}$ , $y = g_{(k-1)L}^{-1} g$ , $h_1 = \Phi_{x}(P)$ , and $J_i = (I_i - (k-1)L)\cap \mathbb{N} = \{j: j\ge 1, j+(k-1)L\in I_i\}$ for $i = 1, 2$ . Then we have $\Phi_g(I_i) = \Phi_x(I) \Phi_y(J_i)$ for $i = 1, 2$ . Thus, $\Phi_y(J_1) = \Phi_y(J_2)$ . By the case of $k=L$ , we have $J_1 \cap \{L\} = J_2 \cap \{L\}$ and the corresponding contracting element is the same. By the definition of $J_i$ , we can conclude that $I_1\cap \{L, \ldots, kL\} = I_2\cap \{L, \ldots, kL\}$ and $f_i = f'_i$ if $a_i\le k$ .

By induction, we conclude that $P_1 = P_2$ and $f_i = f_i'$ for each i. Thus, the lemma holds.

3.6 A gap criterion

In this section, we prove the ‘gap criterion’ giving sufficient conditions on a generating set S and a subset $A \subset G$ to satisfy $\omega_A(d_{S}) \lt \omega_G(d_{S})$ .

The proof uses the critical exponent series of a subset $A\subset G$ , which we define now. Given $g \in G$ , let $\|g\|_S$ denote the word norm of g with respect to the generating set S. For $A \subset G$ and a real parameter $s \geq 0$ , define the Poincaré series of A as

(3.4) \begin{equation} P_A(s) = \sum_{g\in A} e^{-s\|g\|_S}, \end{equation}

which converges for $s \gt \omega_A(d_S)$ and diverges for $s \lt \omega_A(d_S)$ .

Loosely speaking, for a subset $A \subset G$ , the gap criterion provides conditions under which one can construct, for each element $g\in A$ , exponentially many distinct elements in G of word length roughly $\| g\|_S$ . Moreover, for any $g' \in G$ distinct from g, $\mathrm{Im}(\Phi_g) \cap \mathrm{Im}(\Phi_g')$ is disjoint. Since the critical exponent captures exponential growth rates, this yields $\omega_A(S) \lt \omega_G(S)$ .

Fix a finite generating set S of G and denote $\omega_S = \omega_G(d_{S})$ . A similar result appears in [Reference Ding and YangDY24, Lemma 2.26].

1. (a) For any $g\in A$ , there exists an injective map $\Phi_g: \mathcal{P}(n, L) \to G$ where $n = \|g\|_S$ .

2. (b) For any $g\in A$ and $I\in \mathcal{P}(n, L)$ , we have $\|\Phi_g(I)\|_S \le \|g\|_S + |I| (a \log(L) + M)$ .

3. (c) For any $g_1 \neq g_2\in A$ , $\mathrm{Im}(\Phi_{g_1}) \cap \mathrm{Im}(\Phi_{g_2}) = \emptyset$ .

Then $\omega_A(d_{S}) \le \omega_S - ({1}/{L^2})$ .

We note that, in applying this result, the hypothesis that $g \in A$ in condition (1) of Lemma 3.26 will be superfluous, since in §§ 3.4 and 3.5 we have constructed a map $\Phi_g$ satisfying condition (1) of Lemma 3.26 for every $g \in G$ . However, the proof of Lemma 3.26 will only use that the map $\Phi_g$ is defined for $g \in A$ . In applications, the choice of $g \in A$ will play a role in conditions (2) and (3).

Proof. Recall that the critical exponent series of A is defined as

\begin{align*}P_A(s) = \sum_{g\in A} e^{-s\|g\|_S}.\end{align*}

It converges at $s\gt\omega_A(d_{S})$ and diverges at $s\lt\omega_A(d_{S})$ . For any $s\gt\omega_S$ , we have

(3.5)

Denote $F(g) := \sum_{I\in \mathcal{P}(n, L)} (L^{-sa} e^{-sM})^{|I|}$ . Let $\epsilon_i = \chi_I(i)$ for $1\le i\le n$ , $m = \lfloor {n}/{L}\rfloor$ and by the Binomial Theorem we have

\begin{align*} \begin{aligned} F(g) &= \sum_{(\epsilon_1, \ldots, \epsilon_n)\in \mathcal{P}(n, L)} (L^{-sa} e^{-sM})^{\epsilon_1 + \cdots + \epsilon_n}\\ &= (L^{-sa} e^{-sM})^{\lfloor{n}/{L^2}\rfloor} (1+L^{-sa} e^{-sM})^{[{n}/{L}] - [{n}/{L^2}]}\\ &\ge (L^{-sa} e^{-sM})^{{m}/{L}} (1+L^{-sa} e^{-sM})^{m - ({m}/{L})}\\ &= ((L^{-sa} e^{-sM}) (1+L^{-sa} e^{-sM})^{L-1})^{{m}/{L}}, \end{aligned} \end{align*}

where the second equality holds since $\epsilon_i$ must be 1 if $i = kL^2$ and $\epsilon_i$ can be either 0 or 1 otherwise.

Fix any s such that $\omega_S \lt s \lt 2\omega_S$ . Then $sa \lt \frac{1}{4}$ and we have the following inequality:

\begin{align*} \begin{aligned} F(g)^{{L}/{m}} &\ge (L^{-sa} e^{-sM}) (1+L^{-sa} e^{-sM})^{L-1} &\\ &\ge L^{-({1}/{4})} e^{-sM} (1+L^{-({1}/{4})} e^{-sM})^{L-1} & \because\, (1+x)^n \gt nx \\ &\ge L^{-({1}/{4})} e^{-sM} L^{-({1}/{4})} e^{-sM}(L-1) & \because\, L\gt2 \quad\qquad\,\\ &\ge L^{-({1}/{4})} e^{-sM} L^{-({1}/{4})} e^{-sM}{L}/{2} = \tfrac{1}{2} e^{-2sM} L^{{1}/{2}} \ge e, & \end{aligned} \end{align*}

where the last inequality holds since $L^{{1}/{2}} \gt 2 e^{4\omega_S M + 1} \ge 2 e^{2s M + 1}$ . Combined with condition (1), $n = \|g\|_S$ , we obtain

\begin{align*} F(g) \ge e^{{m}/{L}} \ge e^{({n}/{L^2}) - ({1}/{L})}. \end{align*}

Plugging the above inequality into (3.5), we have

\begin{align*} \begin{aligned} P_G(s) &\ge \sum_{g\in A} e^{-s \|g\|_S} e^{({ \|g\|_S}/{L^2}) - ({1}/{L})} = e^{-({1}/{L})} \sum_{g\in A} e^{-(s - ({1}/{L^2})) \|g\|_S} = e^{-({1}/{L})} P_A(s - ({1}/{L^2})). \end{aligned} \end{align*}

Notice that $s \gt \omega_S$ implies $P_G(s)$ converges and $\omega_S \lt s \lt 2\omega_S$ can be chosen arbitrarily. So $P_A(s - ({1}/{L^2}))$ also converges for any $2\omega_S \gt s \gt \omega_S$ . Thus, $\omega_A(d_{S}) \le \omega_S - ({1}/{L^2})$ .

3.7 Approximating by generators

Our goal in this section is to construct the generating sets approximating X, and to show they satisfy the gap criterion, thus proving Theorem C.

Let $\omega_X = \omega_G(d_X)$ be the growth rate of the action. The following result is just a restatement of Lemma 2.1.

We now proceed with the proof of Theorem C′. The main challenge lies in constructing a generating set that is independent of the confined subgroup H. In our approach, we define the generating set as a sufficiently large ball, ensuring that the quasi-geodesic constants (K, C) remain bounded, and in particular K is close to 1. This, in turn, guarantees that the constant a in Lemma 3.14 is sufficiently small, depending only on $\omega_X$ .

For a given confined subgroup H, we utilize the extension map constructed in Definition 3.18. To ensure injectivity in Lemma 3.23, we require the C-contracting axes to be sufficiently long. Specifically, of length greater than $2a \log(L) + D$ , where a is independent of H and D depends on H. Finally, we apply Lemma 3.26 to establish a growth gap.

For convenience, we restate the result here.

Theorem 3.28. Let G act properly coboundedly on a geodesic space (X,d) with a strongly contracting element, and let $x_0 \in X$ . Then there exists $n_0 \gt 0$ such that, for all $n \geq n_0$ , if $S_n$ consists of elements whose $x_0$ -orbit lies in $B(x_0,n) \subset X$ , then $S_n$ generates G and satisfies

\begin{align*}\omega_{G/H}(d_{\bar{S}}) \lt \omega_G(d_S) \end{align*}

for every infinite confined subgroup $H \subset G$ with a non-degenerate confining subset.

Let H be a confined subgroup with a non-degenerate confined subset P. Let A be a set of shortest representatives of $G/H$ : for any $g\in A$ , $\|g\|_S=d_S(1,Hg)$ . Thus, $\omega_{G/H}(\bar{S}) = \omega_A(d_{S})$ .

Let $L_1$ , $L_2$ , D be the constants in Lemmas 3.21 and 3.23.

Let $M = 2(({4D}/{n}) + 1) + \max\{\|p\|_S:p\in P\} + 2\|f\|_S$ , which depends on P (and H). Let $L_3 =L_3(M)$ be given by Lemma 3.26. Choose $L \gt \max\{L_1, L_2, L_3\}$ . To conclude $\omega_{G/H}(d_{\bar{S}}) = \omega_A(d_{S}) \lt \omega_G(d_{S})$ , it suffices to verify the conditions (a), (b), (c) in Lemma 3.26.

By taking a sufficiently large power, we may assume that $2a \log(L) + D \le \|f_i\| \le 2a \log(L) + 2D$ for each $f_i\in \mathcal{F}$ . Thus, for any $g\in A$ , the extension map $\Phi_g: \mathcal{P}(n, L) \to Hg$ defined in Definition 3.18 is injective by Lemma 3.23. This furnishes the injective map in (a). Since $\mathrm{Im}(\Phi_g) \subset Hg$ and A consists of representatives of $G/H$ , we have $\mathrm{Im}(\Phi_{g_1})\cap \mathrm{Im}(\Phi_{g_2}) = \emptyset$ for any $g_1\ne g_2\in A$ , so (c) holds. It remains to prove (b) in what follows.

By Lemma 3.27, we have

\begin{align*}\forall f\in \mathcal F,\quad \|f\|_S \le \frac{1}{n-N} \|f\|_X + 1 \le \frac{2}{n}(2a \log(L) + 2D) + 1,\end{align*}

which yields by the choice of M

\begin{align*}\|f_i f p_i f^{-1} f_i^{-1}\|_S \le 2\|f_i\|_S + 2\|f\|_S + \|p\|_S \le \frac{8a}{n} \log(L) + M.\end{align*}

Notice that

\begin{align*}\frac{8a}{n} \lt \frac{1}{8\omega_X n} \le \frac{1}{8\omega_G(d_{S})}.\end{align*}

Thus, the condition (b) is fulfilled:

\begin{align*}\|\Phi_g(I)\|_S\le \|g\|_S + |I| \bigg(\frac{1}{8\omega_G(d_{S})} \log(L) + M\bigg).\end{align*}

In conclusion, we have verified all the conditions in Lemma 3.26, so it follows that $\omega_{G/H}(d_{\bar{S}}) = \omega_A(d_{S}) \lt \omega_G(d_{S})$ . The proof is complete.

For example, in forthcoming work by the second author (S. Cantrell, D. Martínez-Granado, and E. Reyes, A geometric correspondence for reparameterizations of geodesic flows), Green metrics arising from admissible random walks on G are shown to be dense when G is a non-elementary Gromov hyperbolic group. One might hope to extend this result beyond Gromov hyperbolic groups, at least for injective metrics. Specifically, can every injective metric in the mapping class group be approximated by a sequence of Green metrics?