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Title of Journal: Geom Dedicata

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Abbravation: Geometriae Dedicata

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Springer Netherlands

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10.1002/dmrr.410

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1572-9168

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Extremal behavior of divisibility functions

Authors: Khalid BouRabee D B McReynolds
Publish Date: 2014/01/30
Volume: 175, Issue: 1, Pages: 407-415
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Abstract

In this short article we study the extremal behavior mathrmF Gamma n of divisibility functions mathrmD Gamma introduced by the first author for finitely generated groups Gamma These functions aim at quantifying residual finiteness and bounds give a measurement of the complexity in verifying a word is nontrivial We show that finitely generated subgroups of mathrmGLmK for an infinite field K have at most polynomial growth for the function mathrmF Gamma n Consequently we obtain a dichotomy for the growth rate of log mathrmF Gamma n for finitely generated subgroups of mathrmGLnmathbf C We also show that if mathrmF Gamma n preceq log log n then Gamma is finite In contrast when Gamma contains an element of infinite order log n preceq mathrmF Gamma n We end with a brief discussion of some geometric motivation for this workWe are immensely grateful to the excellent referee for pointing out errors in an earlier draft of this paper We thank Martin Kassabov for asking us a question that led us to find Theorem 13 The first author was partially supported by NSF RTG Grant DMS0602191 The second author was partially supported by NSF DMS1105710

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