The Minimal Growth Rate of Cocompact Coxeter Groups in Hyperbolic 3-space

Abstract

AbstractDue to work of W. Parry it is known that the growth rate of a hyperbolic Coxeter group acting cocompactly on H3 is a Salem number. This being the arithmetic situation, we prove that the simplex group (3,5,3) has the smallest growth rate among all cocompact hyperbolic Coxeter groups, and that it is, as such, unique. Our approach provides a different proof for the analog situation in H2 where E. Hironaka identified Lehmer's number as the minimal growth rate among all cocompact planar hyperbolic Coxeter groups and showed that it is (uniquely) achieved by the Coxeter triangle group (3,7).