# Outer space in Bielefeld

## General Information

The Faculty of Mathematics hosted this conference about outer space, groups of (outer) automorphisms of free groups and related topics. In addition to the talks, there was a session on open problems in this area. This was one of the accompanying workshops of the 50 Years Faculty of Mathematics Anniversary Conference. The conference took place at Bielefeld University.

This conference is supported by the DFG priority programme 2026 Geometry at Infinity

## Speakers

• Yael Algom-Kfir
• Anthony Genevois
• Funda Gultepe
• Camille Horbez
• Marek Kaluba
• Catherine Pfaff
• Dionysios Syrigos

## Programme

TimeTitleSpeaker
Tuesday, 24.09
10:00 - 11:00 Rigidity of the Torelli subgroup in Out(Fn)Camille Horbez
11:30 - 12:30Maximal direct products of free groups in Out(FN)Ric Wade
14:00 - 15:00Typical Trees: An Out(Fr) ExcursionCatherine Pfaff
15:45 - 16:45Problem session
Wednesday, 25.09
10:00 - 11:00The geometry of hyperbolic free-by-cyclic groupsYael Algom-Kfir
11:30 - 12:30The Minimally displaced set of an irreducible automorphism is locally finiteDionysios Syrigos
14:00 - 15:00Homotopy type of the free factor complex Radhika Gupta
15:45 - 16:45Surface subgroups of Out(Fn) via combination of Veech subgroupsFunda Gultepe
19:00 - 00:00Conference Dinner
Thursday, 26.09
10:00 - 11:00Aut(Fn) has property (T)Marek Kaluba
11:30 - 12:30 Graph products, quasi-median graphs, and automorphismsAnthony Genevois

## Abstracts

Yael Algom-Kfir

#### The geometry of hyperbolic free-by-cyclic groups

For $\phi$ an outer automorphism of $F_n$, the corresponding free-by-cyclic group is the group with the following presentation: $G_\phi = \langle x_1, \dots, x_n, t \mid t x_i t^{-1} = \phi(x_i) \rangle$. There is a satisfying correspondence between properties of $\phi$ and properties of the group $G_\phi$. For example, $\phi$ is atoroidal if and only if $G_\phi$ is hyperbolic (Brinkman). In the hyperbolic case, the Gromov boundary of $G_\phi$ contains a cut point if and only if some power of $\phi$ is reducible (Bowditch and Kapovich-Kleiner). We restrict to the case that $G_\phi$ is hyperbolic and $\partial G_\phi$ contains no cut points, then $\partial G_\phi$ is homeomorphic to the Menger curve (Kapovich-Kleiner). Thus, from the point of view of the topology of their boundary, $G_\phi, G_\psi$ are indistinguishable for $\phi \in Out(F_n)$ and $\psi \in Out(F_m)$. However, Paulin showed that the conformal structure of a hyperbolic group is a complete quasi-isometric invariant. Is it possible that all groups of this form are quasi-isometric? In this talk we shall discuss some geometric aspects of $G_\phi$ that effect the conformal structure of $\partial G_\phi$ and ultimately, its dimension. This is joint work with Arnaud Hilion, Emily Stark and Mladen Bestvina.
Anthony Genevois

#### Graph products, quasi-median graphs, and automorphisms

This talk is dedicated to graph products of groups, a common generalisation of free products and right-angled Artin groups. I will explain how quasi-median graphs appear as natural geometric models for graph products, and how they can be used in order to study their automorphisms.
Funda Gultepe

#### Surface subgroups of Out(Fn) via combination of Veech subgroups

Using an amalgamation of Veech subgroups of the mapping class group a la Leininger-Reid, we construct new examples of surface subgroups of $\mathrm{Out}( F_n)$ whose elements are either conjugate to elements in the Veech group, or except for one accidental parabolic, all fully irreducible. I will talk about this construction, and on the way describe Veech subgroups of $\mathrm{Out}(F_n)$ . This is part of a joint work with Binbin Xu.

#### Homotopy type of the free factor complex

The mapping class group of a surface acts on the curve complex which is known to be homotopy equivalent to a wedge of spheres. In this talk, I will define the 'free factor complex', an analog of the curve complex, on which the group of outer automorphisms of a free group acts by isometries. This complex has many similarities with the curve complex. I will present the result (joint with Benjamin Brück) that the free factor complex is also homotopy equivalent to a wedge of spheres. We will also look at higher connectivity results for the simplicial boundary of Outer space.
Camille Horbez

#### Rigidity of the Torelli subgroup in Out(Fn)

I will present a joint work with Sebastian Hensel and Ric Wade. We prove that when n is at least 4, every injective morphism from IAn (outer automorphisms of a free group $F_n$ acting trivially on homology) to $Out(F_n)$ differs from the inclusion by a conjugation. This applies more generally to a wide collection of subgroups of $Out(F_n)$ that we call twist-rich, which include all terms in the Andreadakis-Johnson filtration and all subgroups of $Out(F_n)$ that contain a power of every Dehn twist. This extends previous works on commensurations of $Out(F_n)$ and its subgroups.
Marek Kaluba

#### Aut(Fn) has property (T)

I will sketch the recent proof of (arXiv:1812.03456) that the group of automorphisms of free group on $n \geq 6$ generators has Kazhdan's property (T). The proof follows by estimating the spectral gap of $\Delta_n$, the group Laplace operator via sum of squares decomposition in real group algebra. We use the action of "Weyl" group to simplify the combinatorics of computing $\Delta_n^2$ and reduce the problem of finding a sum of squares decompositions (for all $n \geq 6$) to a single computation for $n=5$. The final computation is just small enough to be performed using computer software. As a side-result we produce asymptotically optimal lower estimates on Kazhdan constants for both $\operatorname{SAut}(F_n)$ and $\operatorname{SL}_n(\mathbb{Z})$. In the latter case these considerably narrow the gap between the upper and lower bounds. This is joint work with Dawid Kielak and Piotr W. Nowak.
Catherine Pfaff

#### Typical Trees: An Out(Fr) Excursion

Random walks are not new to geometric group theory (see, for example, work of Furstenberg, Kaimonovich, Masur). However, following independent proofs by Maher and Rivin that pseudo-Anosovs are generic within mapping class groups, and then new techniques developed by Maher-Tiozzo, Sisto, and others, the field has seen in the past decade a veritable explosion of results. In a 2 paper series, we answer with fine detail a question posed by Handel-Mosher asking about invariants of generic outer automorphisms of free groups and then a question posed by Bestvina as to properties of R-trees of full hitting measure in the boundary of Culler-Vogtmann outer space. This is joint work with Ilya Kapovich, Joseph Maher, and Samuel J. Taylor.
Dionysios Syrigos

#### The Minimally displaced set of an irreducible automorphism is locally finite

Let $G$ be a group which splits as $G = F_n * G_1 *...*G_k$, where every $G_i$ is freely indecomposable and not isomorphic to the group of integers. Guirardel and Levitt generalised the Culler- Vogtmann Outer space of a free group by introducing an Outer space on which $Out(G)$ acts. Francaviglia and Martino introduced the Lipschitz metric for the Culler- Vogtmann space and later for the general Outer space. For any automorphism of $Out(G)$, we can define the displacement function with respect to the Lipschitz metric and the corresponding level sets. Recently, the same authors proved that for every L, the L-level set (the set of points of the outer space which are displaced by at most L) is connected, whenever it is non-empty . In the special case of an irreducible automorphism, they proved that the Min- set (the set of points which are minimally displaced) is always non-empty and it coincides with the set of the points that admit train track representatives for the automorphism. In a joint paper with Francaviglia and Martino, we prove that the Min set of a (hyperbolic) irreducible automorphism is (uniformly) locally finite, even if the relative outer space is locally infinite.

#### Maximal direct products of free groups in Out(FN)

Compared to maximal rank free abelian groups, maximal direct products of free groups in $Out(F_N)$ are remarkably rigid. I will talk about some joint work with Martin Bridson, where we show that every subgroup of $Out(F_N)$ isomorphic to a direct product of 2N-4 free groups fixes a splitting called an N-2 rose. This rose is canonical, so also gives us information about the centralizers and normalizers of such subgroups. As an application, we show that every endomorphism of $Out(F_N)$ sends Nielsen automorphisms to powers of Nielsen automorphisms.

## Participants

• Algom-Kfir, Yael (University of Haifa)
• Beßmann, Lara (University of Münster)
• Brück, Benjamin (Universität Bielefeld)
• Buran, Michal (University of Cambridge)
• Bux, Kai-Uwe (Bielefeld University)
• Carstensen, Thomas (Christian-Albrechts-Universität, Kiel)
• Flechsig, Jonas (Universität Bielefeld)
• Gardam, Giles (University of Muenster)
• Genevois, Anthony (University Paris-Sud)
• Guerch, Yassine (Université Paris-Sud, Laboratoire mathématique d'Orsay)
• Gultepe, Funda (University of Toledo)
• Hilmes, Christoph (Universität Bielefeld)
• Horbez, Camille (CNRS / Université Paris-Sud )
• Kaluba, Marek (Adam Mickiewicz University in Poznań/TU Berlin)
• Kielak, Dawid (Bielefeld University)
• Martino, Armando (Southampton)
• Millard, Ben (University College London)
• Möller, Philip (University of Muenster)
• Neaime, Georges (Universität Bielefeld)
• Pfaff, Catherine (University of California, Santa Barbara)
• Santos Rego, Yuri (OvGU Magdeburg)
• Syrigos, Dionysios (University of Southampton)
• Tielker, Elena (Universität Bielefeld)
• Varghese, Olga (University of Münster)
• Wade, Ric (University of Oxford)
• Witzel, Stefan (Bielefeld University)
• Wu, Xiaolei (University of Bonn)