A class of groups universal for free R-tree actions

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• Thomas Müller, Queen Mary, University of London
• Thursday 14 January 2010, 16:00-17:00
• Watson Building, Lecture Room A.

If you have a question about this talk, please contact Simon Goodwin.

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 X_G, 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|^2ℵ₀,

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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• Algebra Seminar
• School of Mathematics Events
• Watson Building, Lecture Room A

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