PhD Defense Tomasz P. Prytula
Title: Hyperbolic isometries of systolic complexes
The main topics of this thesis are the geometric features of systolic complexes arising from the actions of hyperbolic isometries. Given a hyperbolic isometry h of a systolic complex X, our central theme is to study the minimal displacement set of h and its relation to the actions of h on X and on the systolic boundary of X. We describe the coarse-geometric structure of the minimal displacement set and establish some of its properties that can be seen as a form of quasi-convexity. We apply our results to the study of geometric and algebraic-topological features of systolic groups. In addition, we provide new examples of systolic groups.
In the first part of the thesis we show that the minimal displacement set of a hyperbolic isometry of a systolic complex is quasi-isometric to the product of a tree and the real line. We use this theorem to construct a low-dimensional classifying space for virtually cyclic stabilisers for a group acting properly on a systolic complex, and to describe centralisers of hyperbolic isometries in systolic groups. In the second part of the thesis we are interested in the induced action of h on the systolic boundary, and particularly in the fixed points of this action. The main theorem gives a characterisation of the isometries acting trivially on the boundary in terms of their centralisers in systolic groups.
Jesper Michael Møller, University of Copenhagen
Damian Osajda, University of Wrocław and Polish Academy of Sciences
Nathalie Wahl (chairman), University of Copenhagen
Ian J. Leary, University of Southampton
Jacek Świątkowski, University of Wrocław