Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 13, Number 2 (2013), 1001-1025.
Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot numbers
By a result of R Meyerhoff, it is known that among all cusped hyperbolic 3–orbifolds the quotient of by the tetrahedral Coxeter group has minimal volume. We prove that the group 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 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 is not a Pisot number.
Algebr. Geom. Topol., Volume 13, Number 2 (2013), 1001-1025.
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
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 20F55: Reflection and Coxeter groups [See also 22E40, 51F15]
Secondary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 51F15: Reflection groups, reflection geometries [See also 20H10, 20H15; for Coxeter groups, see 20F55]
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