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Geodesic growth of right-angled Coxeter ...
Ciobanu, Laura...

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Geodesic growth of right-angled Coxeter groups based on trees by Ciobanu, Laura ( Author )
Lemonjar Open Access
N.A
10-04-2015
In this paper we exhibit two infinite families of trees {T1n}n≥17 and {T2n}n≥17 on n vertices, such that T1n and T2n are non-isomorphic, co-spectral, and the right-angled Coxeter groups (RACGs) based on T1n and T2n 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 {S1n}n≥11 and {S2n}n≥11, on n vertices, such that S1n and S2n are non-isomorphic, co-spectral, and the right-angled Coxeter groups (RACGs) based on S1n and S2n have distinct geodesic growth. Asymptotically, as n→∞, each set Tin, or Sin, 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.
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http://arxiv.org/abs/1504.02774
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