## Involve: A Journal of Mathematics

- Involve
- Volume 2, Number 4 (2009), 371-385.

### Automatic growth series for right-angled Coxeter groups

Rebecca Glover and Richard Scott

#### Abstract

Right-angled Coxeter groups have a natural automatic structure induced by their action on a CAT($0$) cube complex. The normal form for this structure is defined with respect to the generating set consisting of all cliques in the defining graph for the group. In this paper we study the growth series for right-angled Coxeter groups with respect to this *automatic *generating set. In particular, we show that there exist nonisomorphic Coxeter groups with the same automatic growth series, and give a comparison with the usual growth series defined with respect to the *standard *generating set.

#### Article information

**Source**

Involve, Volume 2, Number 4 (2009), 371-385.

**Dates**

Received: 11 March 2008

Revised: 8 September 2009

Accepted: 26 September 2009

First available in Project Euclid: 20 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.involve/1513799185

**Digital Object Identifier**

doi:10.2140/involve.2009.2.371

**Mathematical Reviews number (MathSciNet)**

MR2579557

**Zentralblatt MATH identifier**

1223.20030

**Subjects**

Primary: 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 20F10: Word problems, other decision problems, connections with logic and automata [See also 03B25, 03D05, 03D40, 06B25, 08A50, 20M05, 68Q70] 20F55: Reflection and Coxeter groups [See also 22E40, 51F15]

**Keywords**

Coxeter groups growth series

#### Citation

Glover, Rebecca; Scott, Richard. Automatic growth series for right-angled Coxeter groups. Involve 2 (2009), no. 4, 371--385. doi:10.2140/involve.2009.2.371. https://projecteuclid.org/euclid.involve/1513799185