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Seminar by Viveke Erlandsson, Aalto University

Title: Margulis cusps of hyperbolic 4-manifolds

2014.09.09 | Jane Jamshidi

Date Wed 17 Sep
Time 16:15 17:15
Location AUD. D3 (1531-215)


Consider a discrete subgroup G of the isometry group of hyperbolic n-space and a parabolic fixed point p. The Margulis region consists of all points in the space that are moved a small distance by an isometry in the stabilizer of p in G, and is kept precisely invariant under this stabilizer. In dimensions 2 and 3 the Margulis region is always a horoball, which gives the well-understood picture of the
parabolic cusps in the quotient manifold. In higher dimensions, due to the existence of screw-translations (parabolic isometries with a rotational part), this is in no longer true. When the screw-translation has an irrational rotation, the shape of the corresponding region depends on the continued fraction expansion of the irrational angle. In this talk we describe the asymptotic shape of the Margulis region in hyperbolic 4-space corresponding to an irrational screw-translation. Time permitting
we show some consequences to this: that the corresponding parabolic cusps are bi-Lipschitz rigid as well as a necessary discreteness criterion, which can be viewed as a generalization of Shimizu's and Jorgensen's conditions in lower dimensions. This is joint work with Saeed Zakeri.