×

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
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] 10.1090/conm/298/05110
[2] 10.1090/S0002-9939-99-05187-4 · Zbl 0928.52007
[3] 10.1016/j.jfa.2008.10.018 · Zbl 1233.20036
[4] 10.1007/s00039-011-0126-7 · Zbl 1266.20054
[5] 10.1023/A:1021273024728 · Zbl 1025.20027
[6] 10.4213/mzm899 · Zbl 0903.00015
[7] 10.4171/CMH/394 · Zbl 1401.20044
[8] 10.1007/s00039-009-0038-y · Zbl 1207.57005
[9] ; Sageev, Geometric group theory. Geometric group theory. IAS/Park City Math. Ser., 21, 7 (2014)
[10] ; Serre, Arbres, amalgames, SL2. Arbres, amalgames, SL2. Astérisque, 46 (1977)
[11] 10.1007/s00222-003-0321-8 · Zbl 1065.20054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.