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A short proof of a theorem of Brodskii.

James Howie (2000)

Publicacions Matemàtiques

A short proof, using graphs and groupoids, is given of Brodskii’s theorem that torsion-free one-relator groups are locally indicable.

About presentations of braid groups and their generalizations

V. V. Vershinin (2014)

Banach Center Publications

In the paper we give a survey of rather new notions and results which generalize classical ones in the theory of braids. Among such notions are various inverse monoids of partial braids. We also observe presentations different from standard Artin presentation for generalizations of braids. Namely, we consider presentations with small number of generators, Sergiescu graph-presentations and Birman-Ko-Lee presentation. The work of V.~V.~Chaynikov on the word and conjugacy problems for the singular...

Active sums I.

J. Alejandro Díaz-Barriga, Francisco González-Acuña, Francisco Marmolejo, Leopoldo Román (2004)

Revista Matemática Complutense

Given a generating family F of subgroups of a group G closed under conjugation and with partial order compatible with inclusion, a new group S can be constructed, taking into account the multiplication in the subgroups and their mutual actions given by conjugation. The group S is called the active sum of F, has G as a homomorph and is such that S/Z(S) ≅ G/Z(G) where Z denotes the center.The basic question we investigate in this paper is: when is the active sum S of the family F isomorphic to the...

Almost all one-relator groups with at least three generators are residually finite

Mark V. Sapir, Iva Špakulová (2011)

Journal of the European Mathematical Society

We prove that with probability tending to 1, a one-relator group with at least three generators and the relator of length n is residually finite, is a virtually residually (finite p -)group for all sufficiently large p , and is coherent. The proof uses both combinatorial group theory and non-trivial results about Brownian motions.

Amenability of linear-activity automaton groups

Gideon Amir, Omer Angel, Bálint Virág (2013)

Journal of the European Mathematical Society

We prove that every linear-activity automaton group is amenable. The proof is based on showing that a random walk on a specially constructed degree 1 automaton group – the mother group – has asymptotic entropy 0. Our result answers an open question by Nekrashevych in the Kourovka notebook, and gives a partial answer to a question of Sidki.

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