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University of Sydney Algebra Seminar

Anne Thomas

Friday 20 September, 12-1pm, Place: Carslaw 275

The geometry of conjugation in Euclidean isometry groups

We give a simple and beautiful description of the geometry of conjugation within any split subgroup $H$ of the full isometry group $G$ of Euclidean space. We prove that for any $h$ in $H$, the conjugacy class $[h]$ is described geometrically by the move-set of its linearisation, while the set of elements conjugating $h$ to a given $h^{\prime }$ in $[h]$ is described by the fix-set of its linearisation. Examples include affine Coxeter groups, where we give finer results, certain crystallographic groups, and the group G itself. This is joint work with Elizabeth Milićević and Petra Schwer.