## Dr. Eduard Schesler

**Postdoc**

ESI - Erwin Schrödinger International Institute for Mathematics and Physics

E-mail: eduardschesler(at)googlemail.com

Homepage: https://eduardschesler.de/index.html

**Working areas**

Geometric Group Theory

## Project

**58**Profinite perspectives on l2-cohomology

## Publications within SPP2026

We present a new method to construct finitely generated, residually finite, infinite torsion groups. In contrast to known constructions, a profinite perspective enables us to control finite quotients and normal subgroups of these torsion groups. As an application, we describe the first examples of residually finite, hereditarily just-infinite groups with positive first *ℓ*2-Betti-number. In addition, we show that these groups have polynomial normal subgroup growth, which answers a question of Barnea and Schlage-Puchta.

**Related project(s):****58**Profinite perspectives on l2-cohomology

We prove that every finitely generated, residually finite group G embeds into a finitely generated perfect branch group such that many properties of G are preserved under this embedding. Among those are the properties of being torsion, being amenable and not containing a non-abelian free group. As an application, we construct a finitely generated, non-amenable torsion branch group.

Journal | Bull. Lond. Math. Soc. |

Volume | 56 |

Link to preprint version | |

Link to published version |

**Related project(s):****58**Profinite perspectives on l2-cohomology

We prove that the minimal representation dimension of a direct product *G* of non-abelian groups *G*1,…,*G**n* is bounded below by *n*+1 and thereby answer a question of Abért. If each *G**i* is moreover non-solvable, then this lower bound can be improved to be 2*n*. By combining this with results of Pyber, Segal, and Shusterman on the structure of boundedly generated groups we show that branch groups cannot be boundedly generated.

Journal | Arch. Math. |

Volume | 120 |

Link to preprint version | |

Link to published version |

**Related project(s):****58**Profinite perspectives on l2-cohomology

We introduce the notion of a telescope of groups. Very roughly a telescope is a directed system of groups that contains various commuting images of some fixed group *B*. Telescopes are inspired from the theory of groups acting on rooted trees. Imitating known constructions of branch groups, we obtain a number of examples of *B*-telescopes and discuss several applications. We give examples of 2-generated infinite amenable simple groups. We show that every finitely generated residually finite (amenable) group embeds into a finitely generated (amenable) LEF simple group. We construct 2-generated frames in products of finite simple groups and show that there are Grothendieck pairs consisting of amenable groups and groups with property (*τ*). We give examples of automorphisms of finitely generated, residually finite, amenable groups that are not inner, but become inner in the profinite completion. We describe non-elementary amenable examples of finitely generated, residually finite groups all of whose finitely generated subnormal subgroups are direct factors.

**Related project(s):****58**Profinite perspectives on l2-cohomology