arXiv subjects

Stephan Tornier

Publications and source records attributed to Stephan Tornier.

9 recordsLinked to original sources

arXiv2026

The Scale of ($P$)-closed Groups Acting On Trees

Reid--Smith parametrised ($P$)-closed groups acting on trees using graph-based combinatorial structures known as local action diagrams. Properties of the acting (topological) group, such as being locally compact, compactly generated, discrete or simple, are reflected in its local action diagram. In this article, we describe the translations of ($P$)-closed groups and their axes in terms of local action diagrams. As applications, we determine the scale function of ($P$)-closed groups and characterise unimodular as well as uniscalar ($P$)-closed groups. The latter provides one possible answer to a question of Thomas Weigel.

math.GR↗

arXiv2024

Discrete (P)-closed Groups Acting On Trees

Reid-Smith recently parametrised groups acting on trees with Tits' independence property (P) using graph-based combinatorial structures known as local action diagrams. Properties of the acting (topological) group, such as being locally compact, compactly generated or simple, are reflected in its local action diagram. In this article we provide necessary and sufficient conditions on the local action diagram for the associated group to be discrete.

math.GR↗

arXiv2021

Groups Acting on Trees With Prescribed Local Action

We extend Burger--Mozes theory of closed, non-discrete, locally quasiprimitive automorphism groups of locally finite, connected graphs to the semiprimitive case, and develop a generalization of Burger--Mozes universal groups acting on the regular tree $T_{d}$ of degree $d\in\mathbb{N}_{\ge 3}$. Three applications are given: First, we characterize the automorphism types which the quasi-center of a non-discrete subgroup of $\mathrm{Aut}(T_{d})$ may feature in terms of the group's local~action. In doing so, we explicitly construct closed, non-discrete, compactly generated subgroups of $\mathrm{Aut}(T_{d})$ with non-trivial quasi-center, and see that Burger--Mozes theory does not extend further to the transitive case. We then characterize the $(P_{k})$-closures of locally transitive subgroups of $\mathrm{Aut}(T_{d})$ containing an involutive inversion, and thereby partially answer two questions by Banks--Elder--Willis. Finally, we offer a new view on the Weiss conjecture.

math.GR↗

arXiv2021

On free products of graphs

We define a free product of connected simple graphs that is equivalent to several existing definitions when the graphs are vertex-transitive but differs otherwise. The new definition is designed for the automorphism group of the free product to be as large as possible, and we give sufficient criteria for it to be non-discrete. Finally, we transfer Tits' classification of automorphisms of trees and simplicity criterion to free products of graphs.

math.GR↗

arXiv2020

Haar Measures

This article provides a concise introduction to the theory of Haar measures on locally compact Hausdorff groups. We cover the necessary preliminaries on topological groups and measure theory, the Haar correspondence, unimodularity and Haar measures on coset spaces.

math.GR↗

arXiv2020

Prime Localizations of Burger--Mozes-type groups

This article concerns Burger--Mozes universal groups acting on regular trees locally like a given permutation group of finite degree. We also consider locally isomorphic generalizations of the former due to Le Boudec and Lederle. For a large class of such permutation groups and primes $p$ we determine their local $p$-Sylow subgroups as well as subgroups of their $p$-localization, which is identified as a group of the same type in certain cases.

math.GR↗

arXiv2018

Willis Theory via Graphs

We study the scale and tidy subgroups of an endomorphism of a totally disconnected locally compact group using a geometric framework. This leads to new interpretations of tidy subgroups and the scale function. Foremost, we obtain a geometric tidying procedure which applies to endomorphisms as well as a geometric proof of the fact that tidiness is equivalent to being minimizing for a given endomorphism. Our framework also yields an endomorphism version of the Baumgartner-Willis tree representation theorem. We conclude with a construction of new endomorphisms of totally disconnected locally compact groups from old via HNN-extensions.

math.GR↗

arXiv2017

Contraction Groups and Passage to Subgroups and Quotients for Endomorphisms of Totally Disconnected Locally Compact Groups

The concepts of the scale and tidy subgroups for an automorphism of a totally disconnected locally compact group were defined in seminal work by George A. Willis in the 1990s, and recently generalized to the case of endomorphisms (G. A. Willis, Math. Ann. 361 (2015), 403--442). We show that central facts concerning the scale, tidy subgroups, quotients, and contraction groups of automorphisms extend to the case of endomorphisms. In particular, we obtain results concerning the domain of attraction around an invariant closed subgroup.

math.GR↗

arXiv2016

Automorphism Groups of Trees: Generalities and Prescribed Local Actions

This article is an expanded version of the talks given by the authors at the Arbeitsgemeinschaft "Totally Disconnected Groups", held at Oberwolfach in October 2014. We recall the basic theory of automorphisms of trees and Tits' simplicity theorem, and present two constructions of tree groups via local actions with their basic properties: the universal group associated to a finite permutation group by M. Burger and S. Mozes, and the $k$-closures of a given group by C. Banks, M. Elder and G. Willis.

math.GR↗