ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

2. Algebra

1991 MS Classification: 20 Vinberg, Ernest Moscow State University, Russia On a generalization of Coxeter groups Groups defined by relations x_{i}^{k_{i}} = e(1 ≤ i ≤ n), w_{ij}(x_{i}, x_{j})^{k_{ij}} = e(1 ≤ i < j ≤ n), where w_{ij} is a cyclically reduced word in the free product of cyclic groups of orders k_{i} and k_{j}, are considered. It is supposed that k_{i}, k_{ij} ≥ 2 and w_{ij} is not a power of one letter. It is proved that ord x_{i} = k_{i} and ord w_{ij}(x_{i}, x_{j}) = k_{ij}. This extends known results about Coxeter groups and generalized triangle groups. The proof uses ideas of [1] where the partial case w_{ij}(x_{i}, x_{j}) = x_{i}^{p_{ij}}x_{j}^{q_{ij}}, (p_{ij}, k_{i}) = (q_{ij}, k_{j}) = 1, was considered. References:

[1] S. V. Tsaranov, On a generalization of Coxeter groups, Algebras Groups Geom. 6 (1989), 281–318;

[2] E. V. Vinberg, Groups defined by periodic paired relations, Matem. Sbornik 188 (1997), 3–12 (Russian); English transl: Sbornik: Mathematics 188 (1997), 1269–1278.