Speaker: Noam Kolodner (HUJI)
Title: A representation of the outer-automorphisms of the free group

Abstract:
In this talk I will show a construction of an action of the outer-automorphisms of  a free group on a free Z-module that has an ultra-metric. This construction is an attempt to generalize Whitehead's algorithm. Whitehead's algorithm is a classic algorithm in combinatorial group theory used to determine whether a given word is a primitive word i.e. a free generator of the free group. Let w be a cyclic word. In the Whiteheads algorithm one looks at the collection of 2-letter subwords of w to find a whitehead generator of the outer-automorphisms that shortens w. Thus the collection of 2-letter subwords contains information about the action of outer automorphisms on cyclic words. The question I asked myself is what information can be found in the collection of 3-letter subwords of w or the collection of 4-letter subwords of w and so on. One can organize this as a sequence of free modules with projection between them and construct a representation of the outer-automorphisms on the inverse limit. In this representation every outer-automorphism is represented by a tower of finite rectangular matrices.