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Isometry groups of infinite-genus hyperbolic surfaces
Mathematische Annalen ( IF 1.3 ) Pub Date : 2021-03-23 , DOI: 10.1007/s00208-021-02164-z
Tarik Aougab , Priyam Patel , Nicholas G. Vlamis

Given a 2-manifold, a fundamental question to ask is which groups can be realized as the isometry group of a Riemannian metric of constant curvature on the manifold. In this paper, we give a nearly complete classification of such groups for infinite-genus 2-manifolds with no planar ends. Surprisingly, we show there is an uncountable class of such 2-manifolds where every countable group can be realized as an isometry group (namely, those with self-similar end spaces). We apply this result to obtain obstructions to standard group theoretic properties for the groups of homeomorphisms, diffeomorphisms, and the mapping class groups of such 2-manifolds. For example, none of these groups satisfy the Tits Alternative; are coherent; are linear; are cyclically or linearly orderable; or are residually finite. As a second application, we give an algebraic rigidity result for mapping class groups.



中文翻译:

无穷双曲曲面的等距组

给定2个流形,要问的一个基本问题是,可以将哪些组实现为流形上恒定曲率的黎曼度量的等轴测图组。在本文中,我们给出了这类无穷无穷2流形的几乎完全分类。出乎意料的是,我们显示出这类流形中有不可数的一类,其中每个可数组都可以实现为等距组(即那些具有自相似端空间的组)。我们应用该结果来获得对同胚同构,微同同构和此类2流形的映射类组的标准组理论性质的障碍。例如,这些群体都不满足“山雀替代方案”;连贯 是线性的 可循环或线性排序;或者是残差有限的。作为第二个应用,

更新日期:2021-03-23
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