‘Growth’ is a geometrically defined property of a group that can reveal algebraic aspects of the group. For instance, Gromov showed that a group has polynomial growth if and only if it is virtually nilpotent. In this talk, we will focus on growth of groups that act on a CAT(0) cube complex. Such spaces are combinatorial versions of the more general CAT(0) (negatively curved) spaces. For instance, the fundamental group of a closed hyperbolic 3-manifold acts non-trivially on a CAT(0) cube complex. Kar and Sageev showed that if a group acts freely on a CAT(0) square complex, then it either has ‘uniform exponential growth’ or it is virtually abelian. I will present some generalizations of their theorem. This is joint work with Kasia Jankiewicz and Thomas Ng.

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Last updated: 18 May 2024