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Coxeter systems with $2$–dimensional Davis complexes, growth rates and Perron numbers

Naomi Bredon and Tomoshige Yukita

Algebraic & Geometric Topology 24 (2024) 1787–1808
Abstract

We study growth rates of Coxeter systems with Davis complexes of dimension at most 2. We show that if the Euler characteristic χ of the nerve of a Coxeter system is vanishing (resp. positive), then its growth rate is a Salem (resp. Pisot) number. In this way, we extend results due to Floyd (1992) and Parry (1993). In the case where χ is negative, we provide infinitely many nonhyperbolic Coxeter systems whose growth rates are Perron numbers.

Keywords
Coxeter systems, Davis complex, growth rates, Perron numbers
Mathematical Subject Classification
Primary: 20F55, 20F65
References
Publication
Received: 27 December 2022
Revised: 4 February 2023
Accepted: 27 February 2023
Published: 28 June 2024
Authors
Naomi Bredon
Department of Mathematics
University of Fribourg
Fribourg
Switzerland
Tomoshige Yukita
Department of Mathematics
School of Education
Waseda University
Tokyo
Japan

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