## 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.

### MSC:

 20F65 Geometric group theory

### Keywords:

uniform exponential growth; CAT(0) cubical groups
Full Text:

### References:

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