# Third-Party Projects

## Table of contents

## SPP 2026 Geometry at Infinity — Project: Asymptotic geometry of sofic groups and manifolds

**Priority programme of the DFG ** ** ** ** **

Project leader: Dr. Vadim Alekseev,

Prof. Dr. Andreas Thom

Funded by the DFG

Project duration: 04/2017 - 03/2020

#### Proposal summary:

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 Γ 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 Γ-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 Γ 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

#### Positions:

• Dr. Rahel Brugger

• Leonardo Biz

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## Groups, Dynamics, and Approximation

**ERC Consolidator Grant**

Project leader: Prof. Andreas Thom

Funded by the ERC-Consolidator Grant

Project duration: 10/2016 - 09/2021

### Proposal summary:

Eversince, the study of symmetry in mathematics and mathematical physics has been fundamental to a thourough understanding of most of the fundamental notions. Group theory in all its forms is the theory of symmetry and thus an indispensible tool in many of the basic theoretical sciences. The study of infinite symmetry groups is especially challenging, since most of the tools from the sophisticated theory of finite groups break down and new global methods of study have to be found. In that respect, the interaction of group theory and the study of group rings with methods from ring theory, probability, Riemannian geometry, functional analyis, and the theory of dynamical systems has been extremely fruitful in a variety of situations. In this proposal, I want to extend this line of approach and introduce novel approaches to longstanding and fundamental problems.

### Positions:

• Dr. Alessandro Carderi

• Dipl.-Math. Maria Gerasimova

• M.Sc. Jakob Schneider

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## Geometry and Analysis of Group Rings - GeomAnGroup (277728)

**ERC Starting Grant**

Project leader: Prof. Andreas Thom

Funded by the ERC-Starting Grant

Project duration:

10/2011 - 09/2014: U Leipzig,

10/2014 - 09/2016: TU Dresden

### Proposal summary:

Eversince, the study of discrete groups and their group rings has attracted researchers from various mathematical branches and led to beautiful results with proofs involving fields such as number theory, combinatorics and analysis. The basic object of study is the structure of the group G itself, i.e., its subgroups, quotients, etc. and properties of the group ring kG with coefficients in a field k.

Recently, techniques such as Randomization and Algebraic Approximation have lead to fruitful insights. This proposal is focused on new and groundbreaking applications of these two techniques in the study of groups and group rings. In order to illustrate this, I am explaining how useful these techniques are by focusing on three interacting topics:

(i) new characterizations of amenability related to Dixmier’s Conjecture,

(ii) the Atiyah conjecture for discrete groups, and

(iii) algebraic approximation in the algebraic K-theory of algebras of functional analytic type.

All three problems are presently wide open and progress in any of the three problems would mean a breakthrough in current research.

Using Randomization techniques, I want to achieve important results in the understanding of groups rings by contributing to a better understanding of conjectures of Dixmier’s and Atiyah’s. The field of Algebraic Approximation is new, and has already been successfully used by G. Cortinas and myself to resolve a longstanding conjecture in Algebraic K-theory due to Jonathan Rosenberg.

### Positions:

• Dr. Eugene Ha (PostDoc, 12/2011 - 03/2013)

• Dr. Titus Salajan (PostDoc, 04/2013 - 09/2014)

• Dr. Antoine Gournay (PostDoc, 10/2014 - 09/2016)

• Dr. Alessandro Carderi (PostDoc, 09/2015 - 09/2016)

• M.Sc. Christoph Gamm (PhD-student, 11/2011 - 09/2016)

• M.Sc. Marcus de Chiffre (PhD-student, 01/2014 - 09/2016)