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Asymptotic geometry of sofic groups and manifolds

The idea of approximation of infinite structures by means of finite or compact objects is prevalent in modern mathematics. It is the aim of this project to gain insight into the structure of infinite groups and non-compact manifolds by means of sofic approximations.

The project focuses on studying geometric properties for sofic approximations of discrete groups and manifolds. A group $$\Gamma$$ is sofic if it is possible to find local models in symmetric groups $$Sym(X)$$ that are almost multiplicative with respect to a certain metric. Similarly, a cocompact $$\Gamma$$-manifold $$M$$ is called sofic if there exist compact manifolds that locally look like $$M$$ more and more with high probability. A countable collection of finite sets that witness stronger and stronger approximations for an exhaustion of the group $$\Gamma$$ is a sofic approximation - similarly for manifolds.

Sofic groups were introduced by Gromov in his work on Gottschalk's surjunctivity conjecture and later by Weiss. Since then they have played a fundamental role in research in dynamical systems and their connections to $$L^2$$-invariants, notably being the largest class of groups for which the concept of entropy is well defined and the notion of mean dimension has been extended as well as revealing a strong approximation property in the context of $$L^2$$-invariants.

This project consists of three interacting themes:

i) Sofic boundary actions

ii) Study of analytic/geometric properties of groups using actions at infinity

iii) Sofic manifolds

## Publications

We provide a unified treatment of several results concerning full groups of ample groupoids and paradoxical decompositions attached to them. This includes a criterion for the full group of an ample groupoid being amenable as well as comparison of its orbit, Koopman and groupoid-left-regular representations. Besides that, we unify several recent results about paradoxicality in semigroups and groupoids, relating embeddings of Thompson's group V into full groups of ample étale groupoids.

Related project(s):
2Asymptotic geometry of sofic groups and manifolds

We show a rigidity result for subfactors that are normalized by a representation of a lattice Γ in a higher rank simple Lie group with trivial center into a finite factor. This implies that every subfactor of LΓ which is normalized by the natural copy of Γ is trivial or of finite index.

Related project(s):
2Asymptotic geometry of sofic groups and manifolds

We provide a large class of discrete amenable groups for which the complex group ring has several C*-completions, thus providing partial evidence towards a positive answer to a question raised by Rostislav Grigorchuk, Magdalena Musat and Mikael Rørdam.

Related project(s):
2Asymptotic geometry of sofic groups and manifolds

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.

Related project(s):
2Asymptotic geometry of sofic groups and manifolds

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## Team Members

Technische Universität Dresden