Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot numbers

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.