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Groups acting simply transitively on vertex sets of hyperbolic triangular buildings

Part of: Special aspects of infinite or finite groups Algebraic combinatorics

Published online by Cambridge University Press:  01 May 2012

Lisa Carbone ,
Riikka Kangaslampi  and
Alina Vdovina
Lisa Carbone
Affiliation:
Department of Mathematics, Rutgers, The State University of New Jersey, Hill Center, Busch Campus, 110 Frelinghuysen Rd, Piscataway, NJ 08854, USA (email: carbonel@math.rutgers.edu)
Riikka Kangaslampi
Affiliation:
Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, FI-00076 Aalto, Finland (email: riikka.kangaslampi@aalto.fi)
Alina Vdovina
Affiliation:
School of Mathematics and Statistics, University of Newcastle, Newcastle upon Tyne, NE1 7RU, United Kingdom (email: alina.vdovina@ncl.ac.uk)

Abstract

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We construct and classify all groups given by triangular presentations associated to the smallest thick generalized quadrangle that act simply transitively on the vertices of hyperbolic triangular buildings of the smallest non-trivial thickness. Our classification yields 23 non-isomorphic torsion-free groups (which were obtained in an earlier work) and 168 non-isomorphic torsion groups acting on one of two possible buildings with the smallest thick generalized quadrangle as the link of each vertex. In analogy with the case, we find both torsion and torsion-free groups acting on the same building.

MSC classification

Secondary: 05E15: Combinatorial aspects of groups and algebras 20F65: Geometric group theory 20F67: Hyperbolic groups and nonpositively curved groups 05E18: Group actions on combinatorial structures
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 101 - 112
Copyright
Copyright © London Mathematical Society 2012

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