标题： 追踪无限群的有限阴影通过几何学 显示英文标题

标题： Chasing finite shadows of infinite groups through geometry

摘要： 几何学和群论中有许多情形，在这些情形下，研究无限群通过其在有限对象上的作用（即通过群的有限商）是一种自然、方便或者必要的选择。 但是，通过对一个无限群的有限商的研究，我们究竟能够获得多少理解？ 我们可以识别出这个群的哪些性质？什么时候有限商集合能够完全确定这个群？ 判断一个给定的无限群的所有有限商有多困难？ 这些笔记记录了我在2024年7月塞维利亚欧洲数学大会全体会议上的报告内容。 这次报告的目标是概述这些问题丰富的历史，并展示近年来低维拓扑学和非正曲率空间研究的贡献如何彻底改变了围绕这些问题的研究领域。 显示英文摘要

摘要： There are many situations in geometry and group theory where it is natural, convenient or necessary to explore infinite groups via their actions on finite objects, i.e. via the finite quotients of the group. But how much understanding can one really gain about an infinite group by examining its finite images? Which properties of the group can one recognise, and when does the set of finite images determine the group completely? How hard is it to decide what the finite images of a given infinite group are? These notes follow my plenary lecture at the ECM in Sevilla, July 2024. The goal of the lecture was to sketch some of the rich history of the preceding problems and to present results that illustrate how the field surrounding these questions has been transformed in recent years by input from low-dimensional topology and the study of non-positively curved spaces.