#### Volume 13, issue 2 (2013)

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Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot numbers

### Ruth Kellerhals

Algebraic & Geometric Topology 13 (2013) 1001–1025
##### Abstract

By a result of R Meyerhoff, it is known that among all cusped hyperbolic 3–orbifolds the quotient of ${ℍ}^{3}$ by the tetrahedral Coxeter group $\left(3,3,6\right)$ has minimal volume. We prove that the group $\left(3,3,6\right)$ 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 $\left(3,\infty \right)$ 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 $\left(3,3,6\right)$ is not a Pisot number.

##### Keywords
Hyperbolic Coxeter group, cusp, growth rates, Pisot number
##### Mathematical Subject Classification 2000
Primary: 20F55
Secondary: 22E40, 51F15