L²-Invariants for Self-Similar CW-Complexes

by Engelbert Peter Suchla
Doctoral thesis
Date of Examination:2020-10-07
Date of issue:2020-11-11
Advisor:Prof. Dr. Thomas Schick
Referee:Prof. Dr. Dorothea Bahns
Referee:Prof. Dr. Gábor Elek
Persistent Address: http://dx.doi.org/10.53846/goediss-8306

 

 

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Abstract

English

L²-invariants are commonly defined only for periodic spaces, that is, spaces with a cocompact action by some discrete group. This thesis develops a theory of L²-invariants for quasi-periodic spaces instead, where the group action is replaced by a self-similar structure. We show that self-similar CW-complexes have aperiodic order, that is, every geometric pattern of cells appears at a certain well-defined frequency throughout the complex. Using this, we define traces and spectral density functions for geometric operators on the L²-chain groups of such complexes, and we prove that the spectral density functions of such operators can be approximated uniformly by the functions obtained by restricting the operator to finite subcomplexes. We then define L²-Betti numbers, Novikov-Shubin invariants and L²-torsion for self-similar CW-complexes and investigate their most important properties. In particular, we prove that L²-Betti numbers and Novikov-Shubin invariants are indeed invariant under self-similar homotopies, we explore whether they can be approximated by finite-dimensional analogues, and we express the L²-invariants of product spaces in terms of the L²-invariants of their factors.
Keywords: L2-invariants; von Neumann algebras; CW-complexes; self-similarity
 
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