Uniform exponential growth for CAT(0) square complexes. (English) Zbl 07142601

Summary: 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 \(F\) is a finite collection of hyperbolic automorphisms of a CAT(0) square complex \(X\), then either there exists a pair of words of length at most \(10\) in \(F\) which freely generate a free semigroup, or all elements of \(F\) stabilize a flat (of dimension \(1\) or \(2\) in \(X)\). As a corollary, we obtain a lower bound for the growth constant, \(\sqrt[10]{2}\), 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.


20F65 Geometric group theory
Full Text: DOI arXiv


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