Volume preserving subgroups of A and K and singularities in unimodular geometry

三维李群中子群的平均曲率流

The mean curvature flow of subgroups on Lie groups of dimension three

本文研究三维李群中二维李子群的平均曲率流，发现幺模群中李子群为极小曲面，非幺模群中交换子群自相似演化，其他情形则非自相似。

Ends of unimodular random manifolds

We study the ends of a generic manifold, with respect to a unimodular measure on the space of pointed Riemannian manifolds with bounded curvatures. We apply our general result to the case of surfaces and obtain as corollaries a very precise description of generic leaves for foliations with invariant measures and of quotients of the hyperbolic plane by invariant random subgroups of the isometry group.

Unimodular measures on the space of all Riemannian manifolds

We study unimodular measures on the space $\mathcal M^d$ of all pointed Riemannian $d$-manifolds. Examples can be constructed from finite volume manifolds, from measured foliations with Riemannian leaves, and from invariant random subgroups of Lie groups. Unimodularity is preserved under weak* limits, and under certain geometric constraints (e.g. bounded geometry) unimodular measures can be used to compactify sets of finite volume manifolds. One can then understand the geometry of manifolds $M$ with large, finite volume by passing to unimodular limits. We develop a structure theory for unimodu…

Unimodular families of symmetric matrices

We introduce the volume-preserving equivalence among symmetric matrix-valued map-germs which is the unimodular version of Bruce's $\mathcal{G}$-equivalence. The key concept to deduce unimodular classification out of classification relative to $\mathcal{G}$-equivalence is symmetrical quasi-homogeneity, which is a generalization of the condition for a $2 \times 2$ symmetric matrix-valued map-germ in Corollary~2.1 (ii) by Bruce, Goryunov and Zakalyukin. If a $\mathcal{G}$-equivalence class contains a symmetrically quasi-homogeneous representative, the class coincides with that relative to the vol…

The Combinatorial Geometry of Q-Gorenstein Quasi-Homogeneous Surface Singularities

The main result of this paper is a construction of fundamental domains for certain group actions on Lorentz manifolds of constant curvature. We consider the simply connected Lie group G~, the universal cover of the group SU(1,1) of orientation-preserving isometries of the hyperbolic plane. The Killing form on the Lie group G~ gives rise to a bi-invariant Lorentz metric of constant curvature. We consider a discrete subgroup Gamma_1 and a cyclic discrete subgroup Gamma_2 in G~ which satisfy certain conditions. We describe the Lorentz space form Gamma_1\G~/Gamma_2 by constructing a fundamental do…

Cheeger-Chern-Simons classes of representations of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and the spectrum of rational double point singularities

Let $L$ be a compact oriented $3$-manifold and $ρ\colonπ_1(L)\to \mathrm{GL}(n,\mathbb{C})$ a representation. Evaluating the Cheeger-Chern-Simons class $\widehat{c}_{ρ,k}\in H^{2k-1}(L;\mathbb{C}/\mathbb{Z})$ of $ρ$ at $ν\in H_{2k-1}(L;\mathbb{Z})$ we get characteristic numbers that we call the $k$-th CCS-numbers of $ρ$. We prove that if $ρ$ is a topologically trivial representation, the 2-nd CCS-number $\widehat{c}_{ρ,2}([L])$ of the fundamental class $[L]$ of $L$ is given by the invariant $\tildeξ_ρ(D)$ of the Dirac operator $D$ of $L$ twisted by $ρ$ defined by Atiyah, Patodi and Singer. If …