Pure Maths Colloquium: On Margulis cusps of hyperbolic 4-manifolds
26 January 2015 16:00 in CM221
In this talk I will describe cusps in hyperbolic 4-manifolds. In dimension 2 and 3 these cusps are well-understood but in higher dimensions they are in general much more complicated. 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. I will discuss some background in lower dimensions and describe the asymptotic shape of the Margulis region in hyperbolic 4-space. This is joint work with Saeed Zakeri.
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