On Margulis cusps of hyperbolic 4-manifolds

We study the geometry of the Margulis region associated with an irrational screw translation $g$ acting on the 4-dimensional real hyperbolic space. This is an invariant domain with the parabolic fixed point of $g$ on its boundary which plays the role of an invariant horoball for a translation in dimensions $\leq 3$. The boundary of the Margulis region is described in terms of a function $B_\alpha : [0,\infty) \to {\mathbb R}$ which solely depends on the rotation angle $\alpha \in {\mathbb R}/{\mathbb Z}$ of $g$. We obtain an asymptotically universal upper bound for $B_\alpha(r)$ as $r \to \infty$ for arbitrary irrational $\alpha$, as well as lower bounds when $\alpha$ is Diophatine and the optimal bound when $\alpha$ is of bounded type. We investigate the implications of these results for the geometry of Margulis cusps of hyperbolic 4-manifolds that correspond to irrational screw translations acting on the universal cover. Among other things, we prove bi-Lipschitz rigidity of these cusps.Comment: 34 pages, 6 figure

Multiple recurrence for two commuting transformations

This paper is devoted to a study of the multiple recurrence of two commuting transformations. We derive a result which is similar but not identical to that of one single transformation established by Bergelson, Host and Kra. We will use the machinery of "magic systems" established recently by B. Host for the proof

The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space \l^3

We show that a complete embedded maximal surface in the 3-dimensional Lorentz-Minkowski space $L^3$ with a finite number of singularities is, up to a Lorentzian isometry, an entire graph over any spacelike plane asymptotic to a vertical half catenoid or a horizontal plane and with conelike singular points. We study the space $G_n$ of entire maximal graphs over $\{x_3=0\}$ in $L^3$ with $n+1 \geq 2$ conelike singularities and vertical limit normal vector at infinity. We show that $G_n$ is a real analytic manifold of dimension $3n+4,$ and the coordinates are given by the position of the singular points in $R^3$ and the logarithmic growth at the end. We also introduce the moduli space $M_n$ of {\em marked} graphs with $n+1$ singular points (a mark in a graph is an ordering of its singularities), which is a $(n+1)$-sheeted covering of $G_n.$ We prove that identifying marked graphs differing by translations, rotations about a vertical axis, homotheties or symmetries about a horizontal plane, the corresponding quotient space $M_n$ is an analytic manifold of dimension $3n-1.$Comment: 32 pages, 4 figures, corrected typos, former Theorem 3.3 (now Theorem 2.2) modifie