Growth tightness and genericity for word metrics from injective spaces

TL;DR

This paper demonstrates that for mapping class groups acting on injective spaces, large generating sets produce word metrics where pseudo-Anosov maps are exponentially generic, and the groups exhibit growth tightness.

Contribution

It establishes the exponential genericity of pseudo-Anosov maps and growth tightness for certain word metrics derived from injective space actions.

Findings

Pseudo-Anosov maps are exponentially generic in the considered metrics.

Growth tightness holds for the Cayley graphs of these generating sets.

Results provide a positive answer to a question by Arzhantseva, Cashen, and Tao.

Abstract

Mapping class groups are known to admit geometric (proper, cobounded) actions on injective spaces. Starting with such an action, and relying only on geometric arguments, we show that all finite generating sets resulting from taking large enough balls in the respective injective space yield word metrics where pseudo-Anosov maps are exponentially generic. We also show that growth tightness holds true for the Cayley graphs corresponding to these finite generating sets, providing a positive answer to a question by Arzhantseva, Cashen and Tao.