Geodesic growth of right-angled Coxeter groups based on trees

Abstract

In this paper we exhibit two infinite families of trees $\{T_n^1{\}}_{n\ge 17}$ and $\{T_n^2{\}}_{n\ge 17}$ on $n$ vertices, such that $T_n^1$ and $T_n^2$ are non-isomorphic, co-spectral, and the right-angled Coxeter groups (RACGs) based on $T_n^1$ and $T_n^2$ have the same geodesic growth with respect to the standard generating set. We then show that the spectrum of a tree does is not sufficient to determine the geodesic growth of the RACG based on that tree, by providing two infinite families of trees $\{S_n^1{\}}_{n\ge 11}$ and $\{S_n^2{\}}_{n\ge 11}$, on $n$ vertices, such that $S_n^1$ and $S_n^2$ are non-isomorphic, co-spectral, and the right-angled Coxeter groups (RACGs) based on $S_n^1$ and $S_n^2$ have distinct geodesic growth. Asymptotically, as $n\to \infty$, each set $T_n^i$, or $S_n^i$, $i=1,2$, has the cardinality of the set of all trees on $n$ vertices. Our proofs are constructive and use two families of trees previously studied by B. McKay and C. Godsil.

14 pages, 4 figures, a typo in formula (3) corrected; supplementary material and a SAGE worksheet available at http://sashakolpakov.wordpress.com/list-of-papers/