On amenability of two-generator groups

ABSTRACT

A discrete group G is amenable if there exists a finitely additive probability measure on G which is invariant under left translations and is defined on all subsets of G. It is proved that if the group is generated by two elements and is amenable then there are words being relators whose most of the consecutive pairs of the letters belong to a certain four-element set of pairs. This fact is applied to reproving non-amenability of a braid group. The same group provides an example showing that such type of condition is not sufficient for amenabilty.

KEYWORDS:

• Amenable group
• braid group
• Folner sequence
• group presentation

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 20F65
• Secondary 20F36