Naomi Andrew (Université Paris Saclay) - Automorphisms of free-by-cyclic groups
A fundamental question in (geometric) group theory is to determine if a group is finitely generated, presented, or even satisfies some higher finiteness property such as Type VF. I will present new work in this direction for the automorphisms of free-by-cyclic groups, as well as some related questions in Out(Fn). (Joint with Yassine Guerch, Camille Horbez and Ric Wade)
Sasha Bontemps (l'École Normale Supérieure de Lyon) - Subgroup mixing in Baumslag-Solitar groups
Endowed with the Chabauty topology, the space of subgroups of any infinite countable group G is a closed subspace of the Cantor set, equipped with an action by homeomorphisms given by the G-conjugation. We are interested in the dynamics induced by this action. For non-virtually cyclic locally convex hyperbolic groups with trivial finite radical, Hull, Mynasyan and Osin demonstrated strong mixing properties (namely µ-mixing for a suitable measure µ on G, a strengthening of high topological transitivity) on the space of infinite index subgroups. We uncover a radically different situation in the case of non metabelian Baumslag-Solitar groups. For the decomposition of the space of infinite index subgroups introduced by Carderi, Gaboriau, Le Maître and Stalder, who proved high topological transitivity on each piece, we unveil a continuum of measures µ for which the action is µ-mixing only on a single piece of the partition. This yields the first example of a highly topologically transitive action, yet not µ-mixing in the context of groups acting on their space of subgroups.
Montserrat Casals-Ruiz (Euskal Herriko Unibertsitatea) - Automorphisms of graphs of groups with cyclic edge groups
We describe the outer automorphism group of a one-ended fundamental group of a graph of groups, when centralisers of elements in the vertex groups, and the edge groups are infinite cyclic. We show that in this case the outer automorphism group is virtually built from the outer automorphisms of the vertex groups (fixing some elements), the outer automorphisms of some associated generalised Baumslag-Solitar groups, and generalised twists - partial conjugations by elements in the centraliser of some elliptic elements in the group. If time permits, we will relate the isomorphism problem of such graphs of groups with that of generalised Baumslag-Solitar groups. This is joint work with D. Ascari and I. Kazachkov.
Rémi Coulon (Université de Bourgogne) - Power quotients of surface groups and mapping class groups
Given the fundamental group Γ of a surface, we denote by Γ(n) its quotients by the n-th power of every simple closed curve. Such a group appears naturally when studying quantum representations or the quotients of the mapping class group by large powers of Dehn twists.
Nevertheless the nature of Γ(n) as a group is, on the face of it, mysterious: it initially appears to occupy an intermediate status between a surface group and a Burnside group, and might appear to be closer to a Burnside group. In this talk we will present some results suggesting that it is in fact closer to a surface group (e.g. it is virtually torsion free, it fits in a natural Birman-type short exact sequence, etc).
Joint work with A. Sisto and H. Wilton
François Dahmani (l'Université Grenoble Alpes ) - Orbit problems for automorphisms of Dehn-twist mapping tori
Given a free group and two elements the Whitehead orbit problem asks how to know whether they are in the same automorphic orbit. Given graph-of-groups decompositions, this orbit problem and variants appears naturally when one asks whether such or such attaching map is equivalent to another. When vertex groups themselves have splittings, one find oneself considering orbit maps for (unjustly maligned) double cosets of subgroups. When solving conjugacy problem for outer-automorphisms of free groups whose polynomial part is unipotent linear, this orbit problem is important, and can be understood through linkages, subgroups encoding double cosets, after all.
(j.w.w. N. Touikan)
Jerónimo García Mejía (University of Warwick) - Distortion and coarse median preserving automorphisms of special groups
The outer automorphism groups of free groups and mapping class groups sometimes share geometric properties. One such property is that the stable translation length of every infinite-order element is positive with respect to any finite generating set: Farb, Lubotzky, and Minsky established this for mapping class groups in 2001, and Alibegović proved the analogous result for Out(Fn) in 2002. Equivalently, every infinite cyclic subgroup is undistorted.
For a general right-angled Artin group (RAAG) AΓ, however, one cannot expect the same result for the full outer automorphism group, as Out(AΓ) may exhibit behaviour coming from GL(n,Z). In 2016, Charney, Stambaugh, and Vogtmann introduced the subgroup of untwisted automorphisms, which coincides with the whole outer automorphism group when the RAAG is free. Later, in 2024, Fioravanti gave a geometric characterisation of this subgroup as the group of automorphisms preserving a natural coarse median structure on the RAAG.
We prove that every infinite-order element in the group Outcmp(G) of coarse median preserving outer automorphisms has positive stable translation length when G is a RAAG or, more generally, a special group. This is joint work in preparation with Ric Wade.
Yassine Guerch (Université de Caen) - Dynamical invariants of free-by-cyclic groups
A group G is free-by-cyclic if it is isomorphic to the extension of a finitely generated free group F by an infinite cyclic group. The automorphism of F inducing the semi-direct product is not necessarily unique and we want to understand the dynamical properties shared by all such automorphisms.
We in particular study the growth types of the automorphism, which can be described as the possible asymptotic behaviours of iterates (under the automorphism) of elements of F. I will explain how the attracting laminations of the automorphism, which are dynamical objects encoding its exponential growth, induce algebraic invariants of the ambient group G.
Joint work with S. Dowdall, R. Gupta, J.P. Mutanguha and C. Uyanik.
Monika Kudlinska (University of Cambridge) - Thurston norm in groups
The Thurston norm in the setting of 3-manifolds measures the minimal topological complexity of an embedded surface dual to a given integral character. Given a finitely presented group G, it is sometimes possible to define an analogous function on the set of integral characters of G, which measures the minimal complexity of dual HNN splittings over finitely presented subgroups. We prove that if G is free-by-cyclic or the fundamental group of an admissible 3-manifold then such a function extends to a seminorm on the first cohomology of G with real coefficients. To do so, we prove that the complexity function coincides with the twisted L²-Euler characteristic. The main new technical tool is L²-subgroup rigidity for such groups. This is based on joint work with Andrei Jaikin-Zapirain and Pablo Sanchez-Peralta.
Bruno Martelli (Università di Pisa) - A CAT(0) non hyperbolic group without Z x Z
We exhibit a compact 4-dimensional metric space X that is locally CAT(0), and whose fundamental group is not hyperbolic and does not contain Z x Z. The space X has a particularly nice pseudo-Anosov automorphism, that is the main responsible for the non-hyperbolicity of its fundamental group.
Karen Vogtmann (University of Warwick) - Moduli spaces of graphs, surfaces and handlebodies.
Teichmuller space parameterizes marked Riemann surfaces, and Outer space parameterizes marked metric graphs. If you thicken a graph, you get a handlebody, whose boundary is a closed orientable surface. This picture leads to two different notions of a moduli space of marked complex handlebodies, which interpolate between Teichmuller space and Outer space. I will show how augmented versions of these spaces fit together in a nice way. This is joint work with R. Ramadas, R. Silversmith, and R Winarski.