Periodic maximal flats are not peripheral

Peripheral separability and cusps of arithmetic hyperbolic orbifolds

For X = R, C, or H it is well known that cusp cross-sections of finite volume X-hyperbolic (n+1)-orbifolds are flat n-orbifolds or almost flat orbifolds modelled on the (2n+1)-dimensional Heisenberg group N_{2n+1} or the (4n+3)-dimensional quaternionic Heisenberg group N_{4n+3}(H). We give a necessary and sufficient condition for such manifolds to be diffeomorphic to a cusp cross-section of an arithmetic X-hyperbolic (n+1)-orbifold. A principal tool in the proof of this classification theorem is a subgroup separability result which may be of independent interest.

Notes on the peripheral volume of hyperbolic 3-manifolds

We consider hyperbolic 3-manifolds with either non-empty compact geodesic boundary, or some toric cusps, or both. For any such M we analyze what portion of the volume of M can be recovered by inserting in M boundary collars and cusp neighbourhoods with disjoint embedded interiors. Our main result is that this portion can only be maximal in some combinatorially extremal configurations. The techniques we employ are VERY elementary but the result is in our opinion of some interest.

Relatively Hyperbolic Coxeter Groups with maximal Flats of codimension 1

We study relatively hyperbolic Coxeter groups of type $HM$ with maximal Euclidean Coxeter subgroups of codimension 1. Our main result in this paper is that the dimension of these groups is bounded above.

Visual limits of maximal flats in symmetric spaces and Euclidean buildings

Let X be a symmetric space of non-compact type or a locally finite, strongly transitive Euclidean building, and let B denote the geodesic boundary of X. We reduce the study of visual limits of maximal flats in X to the study of limits of apartments in the spherical building B: this defines a natural, geometric compactification of the space of maximal flats of X. We then completely determine the possible degenerations of apartments when X is of rank 1, associated to a classical group of rank 2 or to PGL(4). In particular, we exhibit remarkable behaviours of visual limits of maximal flats in var…

Anosov齐性空间中极大平坦柱面与完整和乐的联合等分布

Joint equidistribution of maximal flat cylinders and holonomies for Anosov homogeneous spaces

本研究针对带极小抛物子群的连通实半单代数群中的Zariski稠密Anosov子群，证明了本原元共轭类对应极大平坦柱面，刻画了周长趋于无穷时柱面与和乐的联合等分布，给出秩一闭测地线相关结果的Anosov类比。

Periodic flats and group actions on locally symmetric spaces

We use maximal periodic flats to show that on a finite volume irreducible locally symmetric manifold of dimension $\geq 3$, no metric $g$ has more symmetry than the locally symmetric metric. We also show that if $g$ is a finite volume metric that is not locally symmetric, then its lift to the universal cover has discrete isometry group.