arXiv subjects

T. Terragni

Publications and source records attributed to T. Terragni.

3 recordsLinked to original sources

arXiv2015

On the growth of a Coxeter group

For a Coxeter system $(W,S)$ let $a_n^{(W,S)}$ be the cardinality of the sphere of radius $n$ in the Cayley graph of $W$ with respect to the standard generating set $S$. It is shown that, if $(W,S)\preceq(W',S')$ then $a_n^{(W,S)}\leq a_n^{(W',S')}$ for all $n\in \mathbb{N}_0$, where $\preceq$ is a suitable partial order on Coxeter systems (cf. Thm. A). It is proven that there exists a constant $τ= 1.13\dots$ such that for any non-affine, non-spherical Coxeter system $(W,S)$ the growth rate $ω(W,S)=\limsup \sqrt[n]{a_n}$ satisfies $ω(W,S)\geq τ$ (cf. Thm. B). The constant $τ$ is a Perron number of degree $127$ over $\mathbb{Q}$. For a Coxeter group $W$ the Coxeter generating set is not unique (up to $W$-conjugacy), but there is a standard procedure, the diagram twisting (cf. [BMMN02]), which allows one to pass from one Coxeter generating set $S$ to another Coxeter generating set $μ(S)$. A generalisation of the diagram twisting is introduced, the mutation, and it is proven that Poincaré series are invariant under mutations (cf. Thm. C).

math.GR↗

arXiv2015

Data about hyperbolic Coxeter systems

We collect several data about Coxeter systems (cf. [Bou07, Hum90]), with particular emphasis on the hyperbolic ones. For each ($\preceq$-minimal) hyperbolic Coxeter system (W,S) the Poincaré series \[p_{(W,S)}(t)=\sum_{w\in W} t^{\ell(w)}\] and the growth rate \[ ω(W,S)=\limsup_n \sqrt[n]{a_n}\] are explicitly computed using Magma (cf. [BCP97]). These computations were performed in connection to the proof of [Ter, Thm. B]. Since the Poincaré series represents a rational function, one may recover the sequence $(a_k)_{k\geq 0}$ through a linear recurrence relation on the coefficients, provided that enough terms at the beginning of the sequence are known. For each Coxeter system the initial coefficients $(a_k)_{k=0}^N$ are computed, where $N$ is the degree of the numerator of $p_{(W,S)}(t)$.

math.GR↗

arXiv2014

The Euler characteristic of a Hecke algebra

It is shown that the Euler characteristic $χ_{(\mathcal{H},\mathcal{B},ε_q)}$ of a $\mathbb{Z}[[q]]$-Hecke algebra $\mathcal{H}$ associated with a finitely generated Coxeter group $(W,S)$ coincides with $p_{(W,S)}(q)^{-1}$, where $p_{(W,S)}(t)$ is the Poincaré series of $(W,S)$.

math.RT↗