Hyperbolicity of one-relator groups and small cancellation techniques

MINIAN, ELIAS GABRIEL

Congreso; Encuentro de Álgebra UBA-CAECE; 2019

Hyperbolic groups were introduced by Gromov in the 80's and they are one of the main objects of study in geometric group theory. A group is hyperbolic if it admits a finite presentation which satisfies an isoperimetric linear inequality. In fact hyperbolicity can be defined in various equivalent ways (I will mention some of these equivalent definitions in the talk). One-relator groups are those which admit presentations with only one relation. It is known that one-relator groups with torsion are hyperbolic, but the geometry of one-relator groups without torsion is more intricated and the general problem is open. The metric small cancellation conditions can be described in terms of the length of the relations and of certain subwords of the relations. It is known that small cancellation conditions C'(1/6) and C'(1/4)-T(4) imply hyperbolicity. Recently in a join work with Martin Blufstein we obtained a metric small cancellation condition that imply hyperbolicity of one-relator groups, more general and flexible than C'(1/6) and C'(1/4)-T(4). In the first part of my talk I will recall  basic results and techniques on hyperbolicity and small cancellation. Then I will describe and motivate some of the ideas behind our recent results in this direction.