According to the Tits conjecture proved by Crisp and Paris, [CP], the subgroups of the braid group generated by proper powers of the Artin elements σi are presented by the commutators of generators which are powers of commuting elements. Hence they are naturally presented as right-angled Artin groups. The case of subgroups generated by powers of the band generators aij is more involved. We show...

In this paper we will present the results of Artin– Markov on braid groups by using the Gröbner-Shirshov basis. As a consequence we can reobtain the normal form of Artin–Markov– Ivanovsky as an easy corollary.

We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group A(K) has such a subgroup if its defining graph K contains an n-hole (i.e. an induced cycle of length n) with n ≥ 5. We construct another eight “forbidden” graphs and show that every graph K on ≤ 8 vertices either contains one of our examples, or co...

In this note we give the quasi-isometry classification for a class of right angled Artin groups. In particular, we obtain the first such classification for a class of Artin groups with dimension larger than 2; our families exist in every dimension.

Given an improper action (= cell stabilizers are innnite) of a group G on a CW-complex X , we present criteria, based on connectivity at innnity properties of the cell stabilizers under the action of G that imply connectivity at innnity properties for G. A reenement of this idea yields information on the topology at innnity of Artin groups, and it gives signiicant progress on the question of wh...

Let p be a prime. Iwasawa’s famous conjecture relating Kubota-Leopoldt p-adic L-functions to the structure of certain Galois groups has been proven by Mazur and Wiles in [10]. Wiles later proved a far-reaching generalization involving p-adic L-functions for Hecke characters of finite order for a totally real number field in [14]. As we discussed in [5], an analogue of Iwasawa’s conjecture for p...

We provide a new presentation for the annular braid group. The annular braid group is known to be isomorphic to the finite type Artin group with Coxeter graph Bn. Using our presentation, we show that the annular braid group is a semidirect product of an infinite cyclic group and the affine Artin group with Coxeter graph Ãn−1. This provides a new example of an infinite type Artin group which inj...

We construct the first examples of normal subgroups mapping class groups that are isomorphic to non-free right-angled Artin groups. Our construction also gives normal, other groups, such as braid and pure well many group, Torelli subgroup. work recovers generalizes seminal result Dahmani–Guirardel–Osin, which free, purely pseudo-Anosov give two applications our methods: (1) we produce an explic...