Boundary Dynamics, Cubical Actions, and Infinite Girth
Tattwamasi Amrutam, Arka Banerjee, Daniel L. Gulbrandsen, Pratyush Mishra
Abstract
We develop two methods for proving infinite girth from boundary dynamics. First, we use topologically free extreme boundary actions to give a boundary-dynamical proof of infinite girth for finitely generated acylindrically hyperbolic groups. The second combines Nakamura's criterion with, respectively, flag-space and Roller-boundary dynamics, yielding infinite-girth results for semisimple S-algebraic groups and for essential non-elementary actions on finite-dimensional, second countable, non-Euclidean CAT(0) cube complexes. We also prove that every finitely generated large group has infinite girth, thereby answering a question of Akhmedov and Mishra.
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