On Growth Functions of Coxeter Groups
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916893257891840 |
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| author | Bischof, Sebastian |
| author_facet | Bischof, Sebastian |
| contents | Let $(W, S)$ be a Coxeter system of rank $n$ and let $p_{(W, S)}(t)$ be its growth function. It is known that $p_{(W, S)}(q^{-1}) < \infty$ holds for all $n \leq q \in \mathbb{N}$. In this paper we will show that this still holds for $q = n-1$, if $(W, S)$ is $2$-spherical. Moreover, we will prove that $p_{(W, S)}(q^{-1}) = \infty$ holds for $q = n-2$, if the Coxeter diagram of $(W, S)$ is the complete graph. These two results provide a complete characterization of the finiteness of the growth function in the case of $2$-spherical Coxeter systems with complete Coxeter diagram. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_10617 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Growth Functions of Coxeter Groups Bischof, Sebastian Combinatorics Group Theory 20F55, 51F15 Let $(W, S)$ be a Coxeter system of rank $n$ and let $p_{(W, S)}(t)$ be its growth function. It is known that $p_{(W, S)}(q^{-1}) < \infty$ holds for all $n \leq q \in \mathbb{N}$. In this paper we will show that this still holds for $q = n-1$, if $(W, S)$ is $2$-spherical. Moreover, we will prove that $p_{(W, S)}(q^{-1}) = \infty$ holds for $q = n-2$, if the Coxeter diagram of $(W, S)$ is the complete graph. These two results provide a complete characterization of the finiteness of the growth function in the case of $2$-spherical Coxeter systems with complete Coxeter diagram. |
| title | On Growth Functions of Coxeter Groups |
| topic | Combinatorics Group Theory 20F55, 51F15 |
| url | https://arxiv.org/abs/2405.10617 |