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A combination theorem for combinatorially non-positively curved complexes of hyperbolic groups

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  09 March 2020

ALEXANDRE MARTIN  and
DAMIAN OSAJDA
ALEXANDRE MARTIN
Affiliation:
Dept. of Mathematics, Heriot-Watt University, EH14 4ASEdinburgh, e-mail: alexandre.martin@hw.ac.uk
DAMIAN OSAJDA
Affiliation:
Instytut Matematyczny, Universytet PL Grunwaldzki 2/4, Wroclawski, 50–384Wroclaw, Poland Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00–656Warszawa, Poland, e-mail: dosaj@math.uni.wroc.pl
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Abstract

We prove a combination theorem for hyperbolic groups, in the case of groups acting on complexes displaying combinatorial features reminiscent of non-positive curvature. Such complexes include for instance weakly systolic complexes and C'(1/6) small cancellation polygonal complexes. Our proof involves constructing a potential Gromov boundary for the resulting groups and analyzing the dynamics of the action on the boundary in order to use Bowditch’s characterisation of hyperbolicity. A key ingredient is the introduction of a combinatorial property that implies a weak form of non-positive curvature, and which holds for large classes of complexes.

As an application, we study the hyperbolicity of groups obtained by small cancellation over a graph of hyperbolic groups.

MSC classification

Primary: 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 20F65: Geometric group theory
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 3 , May 2021 , pp. 445 - 477
Copyright
© Cambridge Philosophical Society 2020

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