The Global Structure of Totally Disconnected Locally Compact Polish Groups

This thesis studies the global structure of totally disconnected locally compact Polish groups. We first identify a fundamental class of totally disconnected locally compact Polish groups: the elementary groups. We go on to prove a variety of local-to-global structure results. In particular, we show the useful and interesting p-adic Lie groups admit decompositions into simple groups and elementary groups via group extension. Our investigations next consider the structure of the collection periodic elements. Here a number of strong partial results are demonstrated. We conclude by studying the structure of totally disconnected locally compact Polish groups with topological conditions on conjugacy classes. Here we answer a question of A. Kechris and C. Rosendal.