# Schedule, Titles & Abstracts

##### Schedule

Time \ Day | Monday | Tuesday | Wednesday | Thursday | Friday |

10:00-10:50 | Manning | Wise | Parlier | Sisto | Webb |

10:50-11:15 | Coffee | Coffee | Coffee | Coffee | Coffee |

11:15-12:05 | Kielak | Hagen | Walsh | Liu | Martin |

12:05-13:40 | Lunch | Lunch | Lunch | Lunch | Lunch |

13:40-14:30 | Osajda | Tao | Free afternoon | Abbott | Woodhouse |

14:40-15:30 | Futer | Jankiewicz | Mangahas | Wilton | |

15:30-16:00 | Tea | Tea | Tea | Tea | |

16:00-16:50 | Mj | Lazarovich | Horbez | ||

18:00 | Dinner | Dinner | Dinner |

##### Titles and abstracts

Title: *Acylindrical actions on hyperbolic spaces*

Abstract: The class of acylindrically hyperbolic groups consists of groups that admit a particular nice type of non-elementary action on a hyperbolic space, called an acylindrical action. This class contains many interesting groups such as non-exceptional mapping class groups, Out(Fn) for n > 1, and right-angled Artin and Coxeter groups, among many others. Such groups admit uncountably many different acylindrical actions on hyperbolic spaces, and one can ask how these actions relate to each other. In this talk, I will describe how to put a partial order on the set of acylindrical actions of a given group on hyperbolic spaces, which roughly corresponds to how much information about the group different actions provide. This partial order organizes these actions into a poset. I will give some structural properties of this poset and, in particular, discuss for which (classes of) groups the poset contains a largest element.

**David Futer** (Temple)

Title: *Generic quotients of cubulated hyperbolic groups*

Abstract: We show that generic quotients of cubulated hyperbolic groups are again cubulated and hyperbolic. Here, “generic” means that the conclusion holds almost surely when long random relations are added at sufficiently low density, in the Gromov density model of random groups.

This is joint work with Dani Wise.

Title: *Guirardel cores for cube complexes*

Abstract: Given a group $G$ acting on two simplicial trees $T_1,T_2$, the Guirardel core is a $G$--invariant subcomplex of $T_1\times T_2$ that records how "compatible" the two splittings of $G$ are, and allows one to sensibly define an intersection number for the two splittings. It turns out that, generalising Guirardel's definition appropriately, one can define a "core" for a finite collection of $G$--actions on CAT(0) cube complexes, and recover some of the same information. The generalisation is especially faithful when $G$ is hyperbolic, although it is also useful when $G$ is, say, a RAAG. I'll discuss some basic properties of this object and an application to the geometry of the free splitting complex of a free group. This talk is mainly on joint work with Henry Wilton and involves some other joint work with Nicholas Touikan.

Title: *Abstract commensurator of some normal subgroups of Out(Fn)*

Abstract: We give a new proof of a theorem of Farb and Handel stating that for all n\ge 4, Out(Fn) is equal to its own abstract commensurator. In other words, every isomorphism between finite-index subgroups of Out(Fn) is the restriction of the conjugation by some element of Out(Fn). Our proof enables us to extend their theorem to the n=3 case. More generally, we prove that many normal subgroups of Out(Fn), for example the Torelli subgroup IA_n and all subgroups in the Johnson filtration of Out(Fn), have Out(Fn) as their abstract commensurator.

This is a joint work with Martin Bridson and Ric Wade.

Title: *Cubical dimension of C’(1/6) groups*

Abstract: The cubical dimension of a group G is the infimum n such that G acts properly on an n dimensional CAT(0) cube complex. For each n we construct examples of C’(1/6) groups with cubical dimension greater than n.

**Dawid Kielak** (Bielefeld)

Title: *Realising groups as mapping tori over surfaces and graphs*

Abstract: We will give a uniform description of all the ways a given 3-manifoldgroup or free-by-cyclic group can be realised as the fundamental group of a mapping torus. In the case of 3-manifolds, such a description was obtained by Thurston, and we will give a new proof of his result.

Title: *Surface-like boundaries of hyperbolic groups*

Abstract: We classify the possible boundaries of hyperbolic groups that have enough quasiconvex codimension-1 surface subgroups with trivial or cyclic intersections.

This is a joint work with Benjamin Beeker.

**Yi Liu** (Beijing)

Title: *Virtual homological spectral radii for automorphisms of surfaces*

Abstract: A surface automorphism is an orientation-preserving self-homeomorphism of a compact orientable surface. A virtual property for a surface automorphism refers to a property which holds up to lifting to some finite covering space. It has been conjectured by C. T. McMullen that any surface automorphism of positive mapping-class entropy possesses a virtual homological eigenvalue which lies outside the unit circle of the complex plane. In this talk, I will review some background and outline a proof of the conjecture.

**Johanna Mangahas** (SUNY, Buffalo)

Title: *Right-angled Artin groups as normal subgroups of mapping class groups*

Abstract: Free normal subgroups of mapping class groups abound, by the result of Dahmani, Guirardel, and Osin that the normal closure of high powers of pseudo-Anosovs is free. At the other extreme, if a normal subgroup contains a mapping class supported on too small a subsurface, it can never be isomorphic to a right-angled Artin group, by work of Brendle and Margalit. I will talk about a case right in between: a family of normal subgroups isomorphic to non-free right-angled Artin groups. We also recover, expand, and make constructive the result of Dahmani, Guirardel, and Osin about free normal subgroups. We do this by creating a version of their “windmill” construction tailor-made for the projection complexes introduced by Bestvina, Bromberg, and Fujiwara.

This is joint work with Matt Clay and Dan Margalit.

Title: *Quasiconvexity and cube complexes*

Abstract: In the 90s, Sageev showed how to build an action on a cube complexfrom a collection of codimension one subgroups of a group G. We consider the case that the group G is hyperbolic, and show that vertex stabilizers are quasiconvex if and only if the codimension one subgroups are. We also give conditions under which the action can be promoted to a proper cocompact action.

This is joint work with Daniel Groves.

**Alexandre Martin** (Heriot-Watt)

Title: *On the automorphism group of graph products of groups*

Abstract: Graph products of groups are a construction that interpolates between direct products and free products, and contain well-known examples such as right-angled Coxeter groups and right-angled Artin groups. In this talk, I will present a form of rigidity for the automorphism group of certain graphs of groups. I will recall a construction due to Davis that allows us to understand graph products through their action on CAT(0) cube complexes, and explain how this action extends to the whole automorphism group in certain cases. Such an action can then be used to completely compute their automorphism group, or to show their acylindrical hyperbolicity.

This is joint work with Anthony Genevois.

Title: *Cubulating surface-by-free groups*

Abstract: Let $1 \to H \to G \to Q \to 1$ be an exact sequence where $H= \pi_1(S)$ is the fundamental group of a closed surface $S$ of genus greater than one, $G$ is hyperbolic and $Q$ is finitely generated free. We shall describe sufficient conditions to prove that $G$ is cubulable. The main result may be thought of as a combination theorem for virtually special hyperbolic groups when the amalgamating subgroup is not quasiconvex. Ingredients include the theory of tracks, the quasiconvex hierarchy theorem of Wise, the distance estimates in the mapping class group from subsurface projections due to Masur-Minsky et al and the model for doubly degenerate Kleinian surface groups used in the proof of the ending lamination theorem.

This is joint work in progress with Jason Manning and Michah Sageev.

Title: *Two-dimensional Artin groups and systolicity*

Abstract: The talk is based on a joint work with Jingyin Huang (MPIM Bonn). We show that two-dimensional Artin groups act geometrically on metrically systolic complexes. The latter are simply connected flag simplicial complexes with links satisfying some non-positive-curvature-like condition. As corollaries we obtain new results on two-dimensional Artin groups and all their finitely presented subgroups: the Dehn function is quadratic, the Conjugacy Problem is solvable. Furthermore, metric systolicity is used by us for studying quasi-isometric rigidity of such groups. In the particular case of large-type Artin groups the results are strengthened - such groups are systolic. In that case we conclude many other new properties, e.g. biautomaticity, existence of an EZ-boundary.

Title: *Simple closed geodesics, hyperbolic surfaces and moduli spaces*

Abstract: Simple closed geodesics on hyperbolic surfaces are related to different aspects of geometric and dynamical properties of moduli spaces and are known to have many remarkable properties. Among these is a result of Birman and Series which states that they form a nowhere dense subset on any closed hyperbolic surface. A particular focus point of the talk will be on joint work with Peter Buser on how to quantify this non-density.

**Alessandro Sisto** (ETH, Zürich)

Title: *Cubulation of hulls of points in hierarchically hyperbolic spaces*

Abstract: It is well-known that the quasi-convex hull of finitely many points in a hyperbolic space is quasi-isometric to a tree. I will discuss an analogous fact in the context of hierarchically hyperbolic spaces, a large class of spaces and groups including mapping class groups, Teichmueller space, RAAGs, RACGs, and others. In this context, the approximating tree is replaced by a CAT(0) cube complex. I will also discuss applications of this cubulation, including a theorem about quasiflats.

Title: *Genus bounds in right-angled Artin groups*

Abstract: I will describe an elementary topological argument which gives lower bounds for stable commutator lengths in right-angled Artin groups.

Title: Relatively hyperbolic groups with Schottky set boundary

Abstract: A Schottky set is the complement in $S^2$ of at least 3 disjoint round open discs. Examples include the Sierpinski carpet and the Apollonian gasket. A relatively hyperbolic group pair $(G, P)$ has a boundary $\partial(G,P)$ which is an invariant of the group pair. When this boundary is a Schottky set, what does this tell us about the group pair? We will show that in this situation the incidence group has 1,2 or infinitely many components and show that in the one-component case, $G$ is virtually a free product of infinite cyclic groups and surface groups.

This is joint work with P. Haissinsky and L. Paoluzzi.

**Richard Webb** (Cambridge)

Title: *How non-positively curved is the mapping class group?*

Abstract: In general, mapping class groups are not CAT(0) but one is still interested in finding non-proper yet cocompact actions on CAT(0) spaces. We will show that, in general, the arc complex admits no CAT(0) metric with finitely many shapes. In particular there is no finite-index subgroup of the mapping class group that preserves a CAT(0) metric on the arc complex. The analogous statements are true for all but finitely many disc complexes of handlebodies and free splitting complexes of free groups.

**Henry Wilton**(Cambridge)

Title: *Surface subgroups of graphs of free groups*

Abstract: Gromov famously asked whether every one-ended hyperbolic group G contains a surface subgroup. I’ll report on recent progress when G is the fundamental group of a graph of free groups.

Title: *No growth-gaps for special cube complexes*

Abstract: It was observed recently with Dahmani and Futer that the growth-rate of an infinite index quasiconvex subgroup H is always less than the growth-rate of G. A celebrated theorem of Corlette states that a quaternionic hyperbolic lattice G has the property that any infinite index subgroup H has growth-rate uniformly bounded away from the growth-rate of G. In contrast to this,

we show that when G is the fundamental group of a compact special cube complex there is a sequence {H_n} of infinite index quasiconvex subgroups whose growth-rates converge to the growth rate of G.

This is joint work with Jiakai Li.

**Daniel Woodhouse** (Technion)

Title: *Leighton's theorem revisited*

Abstract: Leighton's graph covering theorem says that two finite graphs that have common universal cover have a common finite cover. I will present present a new proof that presents the tantalizing possibility of generalizations.