Geodesic growth of right-angled Coxeter groups based on trees
Les taux de croissance pour les groupes de Coxeter rectangulaires basés sur arbres
Résumé
In this paper, we exhibit two infinite families of trees \(\{T^1_n\}_{n \ge 17}\) and \(\{T^2_n\}_{n \ge 17}\) on n vertices, such that \(T^1_n\) and \(T^2_n\) are non-isomorphic, co-spectral, with co-spectral complements, and the right-angled Coxeter groups (RACGs) based on \(T^1_n\) and \(T^2_n\) 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 \(\{S^1_n\}_{n \ge 11}\) and \(\{S^2_n\}_{n \ge 11}\), on n vertices, such that \(S^1_n\) and \(S^2_n\) are non-isomorphic, co-spectral, with co-spectral complements, and the RACGs based on \(S^1_n\) and \(S^2_n\) have distinct geodesic growth. Asymptotically, as \(n\rightarrow \infty \), each set \(T^i_n\), or \(S^i_n\), \(i=1,2\), 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.
Origine : Fichiers produits par l'(les) auteur(s)
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