202223

06 October 2022 at 14:00 in B3.02
Speaker: Grace Garden (University of Sydney)
Title: Earthquakes on the oncepunctured torus
Abstract: We study earthquake deformations on Teichmüller space associated with simple closed curves of the oncepunctured torus. We describe two methods to get an explicit form of the earthquake deformation for any simple closed curve. The first method is rooted in hyperbolic geometry, the second representation theory. The two methods align, providing both a geometric and an algebraic interpretation of the earthquake deformations. Pictures are given for earthquakes across multiple coordinate systems for Teichmüller space. Two families of curves are used as examples. Examining the limiting behaviour of each gives insight into earthquakes about measured geodesic laminations, of which simple closed curves are a special case.

13 October 2022 at 14:00 in B3.02
Speaker: Claudio Llosa Isenrich (KIT)
Title: Finiteness properties, subgroups of hyperbolic groups and complex hyperbolic lattices
Abstract: Hyperbolic groups form an important class of finitely generated groups that has attracted much attention in Geometric Group Theory. We call a group of finiteness type $F_n$ if it has a classifying space with finitely man cells of dimension at most n, generalising finite presentability, which is equivalent to type $F_2$. Hyperbolic groups are of type $F_n$ for all $n$ and it is natural to ask if their subgroups inherit these strong finiteness properties. We use methods from complex geometry to show that every uniform arithmetic lattice with positive first Betti number in $PU(n,1)$ admits a finite index subgroup, which maps onto the integers with kernel of type $F_{n1}$ and not $F_n$. This answers an old question of Brady and produces many finitely presented nonhyperbolic subgroups of hyperbolic groups. This is joint work with Pierre Py.

20 October 2022 at 14:00 in B3.02
Speaker: Henry Bradford (Cambridge)
Title: TBA
Abstract: TBA

27 October 2022 at 14:00 in B3.02
Speaker: Daniel Berlyne (University of Bristol)
Title: Braid groups of graphs
Abstract: The braid group of a space X is the fundamental group of its configuration space, which tracks the motion of some number of particles as they travel through X. When X is a graph, the configuration space turns out to be a special cube complex, in the sense of Haglund and Wise. I show how these cube complexes are constructed and use graph of groups decompositions to provide methods for computing braid groups of various graphs, as well as criteria for a graph braid group to split as a free product. This has various applications, such as characterising various forms of hyperbolicity in graph braid groups and determining when a graph braid group is isomorphic to a rightangled Artin group.

03 November 2022 at 14:00 in B3.02
Speaker: Becca Winarski (College of the Holy Cross)
Title: Polynomials, branched covers, and trees
Abstract: Thurston proved that a postcritically finite branched cover of the plane is either equivalent to a polynomial (that is: conjugate via a mapping class) or it has a topological obstruction. We use topological techniques – adapting tools used to study mapping class groups – to produce an algorithm that determines when a branched cover is equivalent to a polynomial, and if it is, determines which polynomial a topological branched cover is equivalent to. This is joint work with Jim Belk, Justin Lanier, and Dan Margalit.

17 November 2022 at 14:00 in B3.02
Speaker: Bradley Zykoski (Univeristy of Michigan)
Title: A polytopal decomposition of strata of translation surfaces
Abstract: A closed surface can be endowed with a certain locally Euclidean metric structure called a translation surface. Moduli spaces that parametrize such structures are called strata. There is a GL(2,R)action on strata, and orbit closures of this action are rare gems, the classification of which has been given a huge boost in the past decade by landmark results such as the "Magic Wand" theorem of EskinMirzakhaniMohammadi and the Cylinder Deformation theorem of Wright. Investigation of the topology of strata is still in its nascency, although recent work of CalderonSalter and CostantiniMöllerZachhuber indicate that this field is rapidly blossoming. In this talk, I will discuss a way of decomposing strata into finitely many higherdimensional polytopes. I will discuss how I have used this decomposition to study the topology of strata, and my ongoing work using this decomposition to study the orbit closures of the GL(2,R)action.

01 December 2022 at 14:00 in B3.02
Speaker: Koji Fujiwara (Kyoto University)
Title: The rates of growth in a hyperbolic group
Abstract: I discuss the set of rates of growth of a finitely generated group with respect to all its finite generating sets. In a joint work with Sela, for a hyperbolic group, we showed that the set is wellordered, and that each number can be the rate of growth of at most finitely many generating sets up to automorphism of the group. I may discuss its generalization to acylindrically hyperbolic groups.

08 December 2022 at 14:00 in B3.02
Speaker: Ric Wade (University of Oxford)
Title: Autinvariant quasimorphisms on groups
Abstract: For a large class of groups, we exhibit an infinitedimensional space of homogeneous quasimorphisms that are invariant under the action of the automorphism group. This class includes nonelementary hyperbolic groups, infinitelyended finitely generated groups, some relatively hyperbolic groups, and a class of graph products of groups that includes all rightangled Artin and Coxeter groups that are not virtually abelian. Joint work with Francesco FournierFacio.

26 January 2023 at 14:00 in B3.02
Speaker: Davide Spriano (University of Oxford)
Title: Combinatorial criteria for hyperbolicity
Abstract: Perhaps one of the most fascinating properties of hyperbolic groups is that they admit equivalent definitions coming from different areas of mathematics. In this talk, we will survey some interesting definitions, and discuss a new one that, perhaps surprisingly, was previously unknown, namely that fact that hyperbolicity can be detected by the language of quasigeodesics in the Cayley graph. As an application, we will discuss some progress towards a conjecture of Shapiro concerning groups with uniquely geodesic Cayley graphs.

09 March 2023 at 14:00 in D1.07
Speaker: Elia Fioravanti (MPIM Bonn)
Title: Coarse cubical rigidity.
Abstract: When a group G admits nice actions on CAT(0) cube complexes, understanding
the space of all such actions can provide useful information on the outer
automorphism group Out(G). As a classical example, the CullerVogtmann outer
space is (roughly) the space of all geometric actions of the free group F_n on a 1
dimensional cube complex (a tree). In general, however, spaces of cubulations
tend to be awkwardly vast, even for otherwise rigid groups such as the hexagon
RAAG. In an attempt to tame these spaces, we show that all cubulations of many
rightangled Artin and Coxeter groups coarsely look the same, in a strong sense:
they all induce the same coarse median structure on the group. Joint work with
Ivan Levcovitz and Michah Sageev. 
09 March 2023 at 15:00 in D1.07
Speaker: Nansen Petrosyan (Univeristy of Southampton)
Title: Hyperbolicity and Linfinity cohomology
Abstract: Linfinity cohomology is a quasiisometry invariant of finitely generated groups.
It was introduced by Gersten as a tool to find lower bounds for the Dehn
function of some finitely presented groups. I will discuss a generalisation of a
theorem of Gersten on surjectivity of the restriction map in Linfinity
cohomology of groups. This leads to applications on subgroups of hyperbolic
groups, quasiisometric distinction of finitely generated groups and Linfinity
cohomology calculations for some wellknown classes of groups such as RAAGs,
BestvinaBrady groups and Out(F_n). Along the way, we obtain hyperbolicity
criteria for groups of type FP_2(Q) and for those satisfying a rational homological
linear isoperimetric inequality.
I will first define Linfinity cohomology and discuss some of its properties. I will
then sketch some of the main ideas behind the proofs. This is joint work with
Vladimir Vankov.
Seminars from 2022/2023
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