Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot numbers

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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 .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;1/ 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. 20F55; 22E40, 51F15