Geodesic growth of right-angled Coxeter groups based on trees

“And then he had a Clever Idea. He would go up very quietly to the Six Pine Trees ...”

A.A. Milne, Winnie-the-Pooh

Abstract

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.

Appendix

In this section, we give the Python code “Monty” that we used in our computations, with comments and remarks. Its online copy [11] can be downloaded as a SAGE worksheet from the second author’s web-page.

We begin by defining the automaton A that recognises the language of geodesics of a given RACG G whose defining graph $\Gamma$ is given as input.

Given the automaton A, we then compute the growth function of its accepted language (in this case, the geodesic language for G with respect to S).

A clique c in the defining graph $\Gamma$ of the RACG $G = G(\Gamma )$ corresponds to a state $q_c$ in the automaton A. We need the following auxiliary function in order to determine the index of $q_c$ represented as a vertex of the digraph A (the automaton) created by the procedure Automaton.

Below we compute the growth function ${_0}\alpha (t)_G$ as described in the proof of Theorem 1.1.

Now we define a function that takes as input the geodesic automaton A for a RACG $G = G(\Gamma )$, a list of cliques $l = [c_0, c_1, \dots , c_k]$ in the respective defining graph $\Gamma$ and returns the growth function for geodesic words starting at the state $q = \delta (q_0, 0)$ that bring A to any of the states described by the cliques in l.

By using a suitable list of cliques l, we can compute the functions ${_0^0}\alpha (t)_G$ and ${_0^0\beta (t)}_G$. Namely, in the proof of Theorem 1.1, we find

The list l = [[1], [1,3], [1,4], [2], [2,5]] above contains the cliques of $\Gamma _1$ corresponding to the accepting states q of the geodesic automaton $A_1$ for $G_1 = G(\Gamma _1)$ such that $\delta (p, 0) = q$, for a state p. The list l = [[1], [1,3], [2], [2,4], [2,5]] contains the cliques of $\Gamma _2$ with analogous properties, corresponding to the states of the geodesic automaton $A_2$ for the RACG $G_2 = G(\Gamma _2)$.

The above described Python procedures are also used to perform the computations in the proof of Theorem 1.2.

The online version of Monty [11] contains a variation of the GrowthFunc procedure, called GrowthFuncBM, that uses the Berlekamp–Massey algorithm for faster computing.