Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot numbers

Abstract

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 (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,∞) 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.

Growth Rates of 3-dimensional Hyperbolic Coxeter Groups are Perron Numbers

“…Hence if P has a cusp of type (2,3,6) or (2,4,4) or (3, 3, 3), we arrive at a contradiction. This implies that if k = F − 3, P has a unique cusp of type (2, 2, 2, 2) and all other vertices of P are of type (2, 2, m) where m ≥ 7.…”

“…Since P is non-compact, P has at least one cusp. Then if P has a cusp of type (2,3,6) or (2,4,4) or (3,3,3),P also has a cusp of type (2,3,6) or (2,4,4) or (3,3,3). By the fact that a π m -edge (m ≥ 3) is adjacent to two vertices with valency 3, we can see thatP has at least three vertices with valency 3.…”

“…By Theorem 4 and Andreev's theorem [1], if non-compact hyperbolic Coxeter polyhedra with 5 or 6 facets have π m -edges for m ≥ 7, then their combinatorial structures could be (ii), (iv), (v), (viii), (ix), (x). If the combinatorial structure of P is (viii), P has 2 cusps of type (2, 2, 2, 2) and if the combinatorial structure is (ix) or (x), P has at least one of cusps of type (2,3,6) or (2,4,4) or (3,3,3). Hence, the inequality (16) holds for the combinatorial structures (viii), (iv), (x).…”

Hyperbolic Coxeter groups of minimal growth rates in higher dimensions

“…For n = 2, Floyd [6] showed that the Coxeter group Γ 2 = [3, ∞] generated by the reflections in the triangle with angles π/2, π/3 and 0 is the (unique) group of minimal growth rate. For n = 3, Kellerhals [13] proved that the tetrahedral group Γ 3 generated by the reflections in the Coxeter tetrahedron with symbol [6,3,3] realises minimal growth rate in a unique way.…”

“…The Coxeter graphs C 2 and G 2 give rise to the remaining five extensions depicted in Figure 2. By a result of Kellerhals [13], these six Coxeter graphs describe Coxeter tetrahedral groups Λ of finite covolume in IsomH 3 whose growth rates satisfy τ Λ ≥ τ Γ 3 . 1.…”