Margulis cusps and continued fractions
speaker: Saeed Zakeri (City University of New York)abstract: We study the geometry of the Margulis region associated with an irrational screw-translation \( g \) acting on \( {\mathbb H}^4 \). This is an invariant domain with the parabolic fixed point of \( g \) on its boundary which generalizes the notion of an invariant horoball for a translation in dimensions \( \leq 3 \). The boundary of the Margulis region is described by a function \( B_\alpha : [0,\infty) \to {\mathbb R} \) which solely depends on the rotation number \( \alpha \in {\mathbb R}/{\mathbb Z} \) of \( g \). We obtain an asymptotically universal upper bound for \( B_\alpha(r) \) as \( r \to \infty \) for arbitrary \( \alpha \), as well as lower bounds when \( \alpha \) is Diophatine and optimal bounds when \( \alpha \) is of bounded type. We investigate the implications of these results for the geometry of Margulis cusps of hyperbolic \( 4 \)-manifolds. Among other things, we prove that these cusps are bi-Lipschitz rigid.
timetable:
Thu 17 Oct, 10:30 - 11:30, Aula Dini
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