条件表示稳定性、$*$同构的分类和 eta 不变量

Rufus Willett
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引用次数: 0

摘要

一个群的准表示是一个从群映射到矩阵代数(或类似对象)的映射,它近似满足表示所需的关系。从卡兹丹(Kazhdan)和沃伊库列斯库(Voiculescu)开始,以及最近由达达尔拉特(Dadarlat)、埃勒斯-舒尔曼-索伦森(Eilers-Shulman-S\o{}rensen)等人所做的工作表明,用诚实表示来近似群的单元准表示存在拓扑障碍,这里的 "近似 "是指关于算子规范的近似。本文的目的是探讨如果已知的障碍消失,近似是否可能。本文部分推广了 Gong-Lin 和 Eilers-Loring-Pedersen 针对二阶自由无性群和克莱因波特群所做的工作。我们证明,至少在弱意义上,这对一些 "低维 "群是可能的,包括封闭曲面的基群、某些鲍姆斯拉格-索利特群、自由逐周期群和许多三流形的基群。论文中使用的技术是 K$ 理论的:它们起源于鲍姆-康内斯-卡斯帕罗夫类型的集合映射,以及埃利奥特的 C^*$ 矩阵分类计划;卡斯帕罗夫的双变量 KK 理论是一个关键工具。关键的新技术成分是:达达拉特-埃勒斯和林的意义上的稳定唯一性定理,该定理适用于非act $C^*$-gebras;以及根据阿蒂亚-帕托迪-辛格的等效不变式分析 $K$-theory 上具有有限系数的映射。尽管这些证明都要通过K$理论机制,但主要定理可以用不需要任何K$理论的基本定理来表述。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Conditional representation stability, classification of $*$-homomorphisms, and eta invariants
A quasi-representation of a group is a map from the group into a matrix algebra (or similar object) that approximately satisfies the relations needed to be a representation. Work of many people starting with Kazhdan and Voiculescu, and recently advanced by Dadarlat, Eilers-Shulman-S\o{}rensen and others, has shown that there are topological obstructions to approximating unitary quasi-representations of groups by honest representations, where `approximation' is understood to be with respect to the operator norm. The purpose of this paper is to explore whether approximation is possible if the known obstructions vanish, partially generalizing work of Gong-Lin and Eilers-Loring-Pedersen for the free abelian group of rank two, and the Klein bottle group. We show that this is possible, at least in a weak sense, for some `low-dimensional' groups including fundamental groups of closed surfaces, certain Baumslag-Solitar groups, free-by-cyclic groups, and many fundamental groups of three manifolds. The techniques used in the paper are $K$-theoretic: they have their origin in Baum-Connes-Kasparov type assembly maps, and in the Elliott program to classify $C^*$-algebras; Kasparov's bivariant KK-theory is a crucial tool. The key new technical ingredients are: a stable uniqueness theorem in the sense of Dadarlat-Eilers and Lin that works for non-exact $C^*$-algebras; and an analysis of maps on $K$-theory with finite coefficients in terms of the relative eta invariants of Atiyah-Patodi-Singer. Despite the proofs going through $K$-theoretic machinery, the main theorems can be stated in elementary terms that do not need any $K$-theory.
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