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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 \(h'\) 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.