PhD Defense Tomasz P. Prytula – University of Copenhagen

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PhD Defense Tomasz P. Prytula

Title: Hyperbolic isometries of systolic complexes

Abstract:

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.

Supervisors:  

Jesper Michael Møller, University of Copenhagen

Damian Osajda, University of Wrocław and Polish Academy of Sciences

Assessment committee:

Nathalie Wahl (chairman), University of Copenhagen

Ian J. Leary, University of Southampton

Jacek Świątkowski, University of Wrocław