We start the inquiry into proving uniform exponential growth in the context of groups acting on CAT(0) cube complexes. We address free group actions on CAT(0) square complexes and prove a more general statement. This says that if is a finite collection of hyperbolic automorphisms of a CAT(0) square complex , then either there exists a pair of words of length at most in which freely generate a free semigroup, or all elements of stabilize a flat (of dimension or in ). As a corollary, we obtain a lower bound for the growth constant, , which is uniform not just for a given group acting freely on a given CAT(0) cube complex, but for all groups which are not virtually abelian and have a free action on a CAT(0) square complex.
"Uniform exponential growth for CAT(0) square complexes." Algebr. Geom. Topol. 19 (3) 1229 - 1245, 2019. https://doi.org/10.2140/agt.2019.19.1229