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.

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，