14.00 – 14.45   Elia Fioravanti (KIT – Karlsruhe) « Generators for automorphisms of special groups »
15.00 – 15.45   Enrico Le Donne (University of Fribourg) « Asymptotic geometry of Riemannian nilpotent groups »
16.15 – 17.00   Catherine Pfaff (Queen’s University) « A « cubist » decomposition of the Handel-Mosher axis bundle »

Abstracts:

Elia Fioravanti: Generators for automorphisms of special groups 

Given a family F of finitely generated groups, do all groups in F have “tame” automorphisms, or can there be “wild” examples? More concretely, is Out(G) finitely generated for all groups G in the family F? Rips and Sela showed in the 90s that Out(G) is finitely generated for all Gromov-hyperbolic groups G, while Baues and Grunewald showed in the 00s that Out(G) is arithmetic over Q (and hence finitely generated) for all virtually polycyclic groups G. This essentially exhausts our limited understanding of general phenomena of this kind, with the structure of automorphisms of non-positively curved groups remaining a fundamental open problem. I will discuss the recent result that Out(G) is finitely generated for all (cocompact) special groups of Haglund and Wise. This is already new for most finite-index subgroups of right-angled Artin and Coxeter groups.

Enrico Le Donne: Asymptotic geometry of Riemannian nilpotent groups.

Asymptotic cones of Riemannian nilpotent Lie groups are Carnot groups. The volume of balls in Carnot groups grows exactly as a power of the radius. Heuristically, the better the asymptotic cone approximates a Riemannian group, the closer the volume growth approaches a polynomial growth. I will discuss several results obtained over the last few years in collaboration with Breuillard, Nalon, Nicolussi Golo, and Ryoo.

Catherine Pfaff: A “cubist” decomposition of the Handel-Mosher axis bundle

A hyperbolic isometry acts on the compactified hyperbolic plane with North-South dynamics and a single invariant axis. The same is true for a pseudo-Anosov mapping class acting on a Teichmuller space and other hyperbolic-like settings. However, while a fully irreducible free group outer automorphism acts on compactified Outer space with North-South dynamics, there can be many axes for a single fully irreducible φ ∈ Out(F_r). With this in mind, Handel and Mosher define the axis bundle for a fully irreducible φ ∈ Out(F_r). And then Handel-Mosher and Bridson-Vogtmann ask about the geometry of the axis bundle. In joint work with Chi Cheuk Tsang, we show that the axis bundle of a nongeometric fully irreducible outer automorphism admits a canonical “cubist” decomposition into branched cubes that fit together with special combinatorics. From this structure, we locate a canonical finite collection of periodic fold lines in each axis bundle. This can be considered as an analogue of results of Hamenstadt and Agol from the surface setting, which state that the set of trivalent train tracks carrying the unstable lamination of a pseudo-Anosov map can be given the structure of a CAT(0) cube complex, and that there is a canonical periodic fold line in this cube complex. Our “cubist” decomposition also gives a “hands on” solution to the fully irreducible