University of Birmingham > Talks@bham > Algebra Seminar > A class of groups universal for free R-tree actions

A class of groups universal for free R-tree actions

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I report on a new construction in group theory giving rise to a kind of continuous analogue of free groups. More explicitly, given any (discrete) group G, we construct a group RF(G) equipped with a natural (real-valued) Lyndon length function, and thus with a canonical action on an associated R-tree XG, which turns out to be transitive. Analysis of these groups RF(G) is difficult. However, conjugacy of hyperbolic elements is understood, as are the centralizers and normalizers of hyperbolic elements; moreover, we show that RF-groups and their associated R-trees are universal (with respect to inclusion) for free R-tree actions. Furthermore, we prove that

|RF(G)| = |G|20,

and that non-trivial normal subgroups of RF(G) contain a free subgroup of rank |RF(G)|, as well as a number of further structural properties of RF(G) and its quotient by the span of the elliptic elements.

This talk is part of the Algebra Seminar series.

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