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Divergence and quasi-isometry classes of random Gromov’s monsters

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  18 February 2021

DOMINIK GRUBER  and
ALESSANDRO SISTO
DOMINIK GRUBER
Affiliation:
Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland. e-mail: dom.gruber@gmx.at
ALESSANDRO SISTO
Affiliation:
Department of Mathematics, Heriot-Watt University, Edinburgh e-mail: a.sisto@hw.ac.uk
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Abstract

We show that Gromov’s monsters arising from i.i.d. random labellings of expanders (that we call random Gromov’s monsters) have linear divergence along a subsequence, so that in particular they do not contain Morse quasigeodesics, and they are not quasi-isometric to Gromov’s monsters arising from graphical small cancellation labellings of expanders.

Moreover, by further studying the divergence function, we show that there are uncountably many quasi-isometry classes of random Gromov’s monsters.

MSC classification

Primary: 20F65: Geometric group theory
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 2 , September 2021 , pp. 249 - 264
Copyright
© Cambridge Philosophical Society 2021

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