## Algebraic & Geometric Topology

### Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot numbers

Ruth Kellerhals

#### Abstract

By a result of R Meyerhoff, it is known that among all cusped hyperbolic 3–orbifolds the quotient of $ℍ3$ by the tetrahedral Coxeter group $(3,3,6)$ has minimal volume. We prove that the group $(3,3,6)$ has smallest growth rate among all non-cocompact cofinite hyperbolic Coxeter groups, and that it is as such unique. This result extends to three dimensions some work of W Floyd who showed that the Coxeter triangle group $(3,∞)$ has minimal growth rate among all non-cocompact cofinite planar hyperbolic Coxeter groups. In contrast to Floyd’s result, the growth rate of the tetrahedral group $(3,3,6)$ is not a Pisot number.

#### Article information

Source
Algebr. Geom. Topol., Volume 13, Number 2 (2013), 1001-1025.

Dates
Received: 12 July 2012
Revised: 30 November 2012
Accepted: 5 December 2012
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715545

Digital Object Identifier
doi:10.2140/agt.2013.13.1001

Mathematical Reviews number (MathSciNet)
MR3044599

Zentralblatt MATH identifier
1281.20044

#### Citation

Kellerhals, Ruth. Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot numbers. Algebr. Geom. Topol. 13 (2013), no. 2, 1001--1025. doi:10.2140/agt.2013.13.1001. https://projecteuclid.org/euclid.agt/1513715545

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