Geodesic growth of right-angled Coxeter groups based on trees

Abstract

In this paper, we exhibit two infinite families of trees $$\{T^1_n\}_{n \ge 17}$${Tn1}nź17 and $$\{T^2_n\}_{n \ge 17}$${Tn2}nź17 on n vertices, such that $$T^1_n$$Tn1 and $$T^2_n$$Tn2 are non-isomorphic, co-spectral, with co-spectral complements, and the right-angled Coxeter groups (RACGs) based on $$T^1_n$$Tn1 and $$T^2_n$$Tn2 have the same geodesic growth with respect to the standard generating set. We then show that the spectrum of a tree is not sufficient to determine the geodesic growth of the RACG based on that tree, by providing two infinite families of trees $$\{S^1_n\}_{n \ge 11}$${Sn1}nź11 and $$\{S^2_n\}_{n \ge 11}$${Sn2}nź11, on n vertices, such that $$S^1_n$$Sn1 and $$S^2_n$$Sn2 are non-isomorphic, co-spectral, with co-spectral complements, and the RACGs based on $$S^1_n$$Sn1 and $$S^2_n$$Sn2 have distinct geodesic growth. Asymptotically, as $$n\rightarrow \infty $$nźź, each set $$T^i_n$$Tni, or $$S^i_n$$Sni, $$i=1,2$$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.