Characters on infinite groups and rigidity

by Rahel Brugger
Doctoral thesis
Date of Examination:2018-02-07
Date of issue:2018-05-02
Advisor:Prof. Dr. Thomas Schick
Referee:Prof. Dr. Thomas Schick
Referee:Prof. Dr. Ralf Meyer
Persistent Address: http://dx.doi.org/10.53846/goediss-6832

 

 

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Abstract

English

We show that for a strong extension of discrete measured groupoids $1\to\mathcal{S}\to\mathcal{G}\to\mathcal{Q}\to 1$ with $L\mathcal{G}$ a finite factor, $\mathcal{Q}$ has poperty (T) if and only if the inclusion of $L\mathcal{S}$ into $L\mathcal{G}$ is corigid. In particular, this implies that $\mathcal{G}$ has property (T) if and only if $L^\infty(X)\subset L\mathcal{G}$ is corigid. Furthermore, we give the definition of an invariant random positive definite function on a discrete group, generalizing both the notion of an Invariant Random Subgroup and a character. We use von Neumann algebras to show that all invariant random positive definite functions on groups with infinite conjugacy classes which integrate to the regular character are constant. We also show a rigidity result for subfactors that are normalized by a representation of a lattice $\Gamma$ in a higher rank simple Lie group with trivial center into a finite factor. This implies that every subfactor of $L\Gamma$ which is normalized by the natural copy of $\Gamma$ is trivial or of finite index.
Keywords: von Neumann algebras; property (T); invariant random positive definite functions; measured groupoids; characters; rigidity
 
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