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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.

Almost finitely presented soluble groups.

Robert Bieri, Ralph Strebel (1978)

Commentarii mathematici Helvetici

Almost fixed-point-free automorphisms of prime order

Bertram Wehrfritz (2011)

Open Mathematics

Let ϕ be an automorphism of prime order p of the group G with C G(ϕ) finite of order n. We prove the following. If G is soluble of finite rank, then G has a nilpotent characteristic subgroup of finite index and class bounded in terms of p only. If G is a group with finite Hirsch number h, then G has a soluble characteristic subgroup of finite index in G with derived length bounded in terms of p and n only and a soluble characteristic subgroup of finite index in G whose index and derived length are...

Amalgamated Free Products of Groups and Homological Duality

Robert Bieri, Benno Eckmann (1974)

Commentarii mathematici Helvetici

Amalgamated products and finitely presented groups.

Gilbert Baumslag, Peter B. Shalen (1990)

Commentarii mathematici Helvetici

Amalgams of torsion-free nilpotent groups of class three

Berthold J. Maier (1984)

Rendiconti del Seminario Matematico della Università di Padova

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.

Amenable actions of amalgamated free products of free groups over a cyclic subgroup and generic property

Soyoung Moon (2011)

Annales mathématiques Blaise Pascal

We show that the amalgamated free products of two free groups over a cyclic subgroup admit amenable, faithful and transitive actions on infinite countable sets. This work generalizes the results on such actions for doubles of free group on two generators.

Amenable actions of nonamenable groups.

Grigorchuk, R.I., Nekrashevych, V.Yu. (2005)

Zapiski Nauchnykh Seminarov POMI

Amenable hyperbolic groups

Pierre-Emmanuel Caprace, Yves de Cornulier, Nicolas Monod, Romain Tessera (2015)

Journal of the European Mathematical Society

We give a complete characterization of the locally compact groups that are non elementary Gromov-hyperbolic and amenable. They coincide with the class of mapping tori of discrete or continuous one-parameter groups of compacting automorphisms. We moreover give a description of all Gromov-hyperbolic locally compact groups with a cocompact amenable subgroup: modulo a compact normal subgroup, these turn out to be either rank one simple Lie groups, or automorphism groups of semiregular trees acting doubly...

Amenable, transitive and faithful actions of groups acting on trees

Pierre Fima (2014)

Annales de l’institut Fourier

We study under which condition an amalgamated free product or an HNN-extension over a finite subgroup admits an amenable, transitive and faithful action on an infinite countable set. We show that such an action exists if the initial groups admit an amenable and almost free action with infinite orbits (e.g. virtually free groups or infinite amenable groups). Our result relies on the Baire category Theorem. We extend the result to groups acting on trees.

An acyclic extension of the braid group.

Vlad Sergiescu, Peter Greenberg (1991)

Commentarii mathematici Helvetici

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