Counting pseudo-Anosovs as weakly contracting isometries

TL;DR

This work proves that pseudo-Anosov elements are generic in every Cayley graph of a mapping class group Mod$(\Sigma)$, with non-pseudo-Anosov density decaying at least as $K/\sqrt{R}$ and stronger superpolynomial rates. The key innovation is a framework built from weakly contracting geodesics, WPD, and alignment in the curve complex, coupled with concatenation lemmas that control how φ-orbit segments contribute to word length growth. The results extend to rank-1 CAT(0) groups and hierarchically hyperbolic groups with Morse elements, and, via recent quasi-isometric rigidity, to groups quasi-isometric to well-behaved HHGs, yielding a quasi-isometry invariant counting theory for Morse/weakly contracting elements. A suite of applications follows, including genericity statements for Morse elements in braid groups and an overarching approach that applies beyond relatively hyperbolic settings to a broad class of groups acting on contracting spaces. Overall, the paper provides a robust, QI-invariant toolbox for counting hyperbolic-like elements across diverse geometric group-theoretic contexts.

Abstract

We show that pseudo-Anosov mapping classes are generic in every Cayley graph of the mapping class group of a finite-type hyperbolic surface. Our method also yields an analogous result for rank-one CAT(0) groups and hierarchically hyperbolic groups with Morse elements. Finally, we prove that Morse elements are generic in every Cayley graph of groups that are quasi-isometric to (well-behaved) hierarchically hyperbolic groups. This gives a quasi-isometry invariant theory of counting group elements in groups beyond relatively hyperbolic groups.

Figures (4)

• Figure 1: Schematics for alignment in $\mathcal{C}(\Sigma)$. In this picture, the sequence of geodesics $(\gamma_{0}, \gamma_{1}, \ldots, \gamma_{5})$ is aligned, where $\gamma_{0}, \gamma_{5}$ are degenerate geodesics, i.e., points.
• Figure 2: Schematics for Proposition \ref{['prop:weakConcat2']}. When $(gx_{0}, \mathop{\mathrm{\operatorname{Proj}}}\nolimits \gamma_{1}, \ldots, \mathop{\mathrm{\operatorname{Proj}}}\nolimits \gamma_{n}, hx_{0})$ is aligned, some part of $[g, h]_{S}$ is brought close to $\gamma_{i}$'s. Note that the contribution of $d_{S}(\gamma_{i}, \gamma_{i+1})$ to the threshold of $d_{S}([g, h]_{S}, \gamma_{j})$ decays exponentially in $|i-j|$.
• Figure 3: Schematics for Corollary \ref{['cor:weakConcatSqrt']} in the case $n=4$. The alignment forces that $\gamma_{i}$ moves away from $g$ (in $d_{S}$) at least linearly fast. Since $[h_{1}, h_{2}]_{S}$ is a geodesic, $D_{i} + D_{i}'$ cannot be small and grows linearly as well.
• Figure 4: Construction of $F_{n}(g, i)$. The upper layer is drawn on the Cayley graph of $\mathop{\mathrm{Mod}}\nolimits(\Sigma)$, while the lower one is drawn on the curve complex $\mathcal{C}(\Sigma)$. The words $w(g, i)$, $l(g, i)$ and $v(g, i)$ are defined by chopping up a $d_{S}$-geodesic between $id$ and $g$ at suitable loci. Here, $x_{0}, wx_{0}, wlx_{0}, wlvx_{0} = gx_{0}$ may not be aligned along a geodesic on $\mathcal{C}(\Sigma)$. This can be remedied by replacing $l$ with some suitable linkage word $s \varphi^{2L_{map}}t$, and we denote the resulting product by $F_{n}(g, i)$. Note that $ws[x_{0}, \varphi^{2L_{map}} x_{0}]_{\mathcal{C}}$ is uniformly close to $[x_{0}, F_{n}(g, i)x_{0}]_{\mathcal{C}}$.

Theorems & Definitions (38)

• Conjecture 1.1
• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Definition 2.1: Weak proper discontinuity, bestvina2002bounded
• proof : Sketch of proof
• proof
• Definition 3.2
• proof