Please contact Saul Schleimer or Robert Kropholler if you would like to speak or to suggest a speaker.
While this page is the seminar's "main page", I will attempt to also maintain an up-to-date listing at researchseminars.org.
The seminar will be hybrid, and will be run weekly. The talk is in B3.02 Zeeman Building on Thursdays, starting at 14:05. We will open and close the Zoom session on the hour. Note that no password is required; links to the zoom session for each talk are below.
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Thursday October 6, 14:05 (UK time), B3.02 Zeeman. Grace Garden (Sydney) Earthquakes on the once-punctured torus |
Abstract: We study earthquake deformations on Teichmüller space associated with simple closed curves of the once-punctured 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. |
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Thursday October 13, 14:05 (UK time), B3.02 Zeeman. Claudio Llosa Isenrich (KIT) 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 many cells of dimension at most \(n\). This generalises finite presentability, which is equivalent to type \(F_2\). Hyperbolic groups are of type \(F_n\) for all \(n\). It is natural to ask if subgroups of hyperbolic groups 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 \(F_{n−1}\) but not \(F_n\). This answers an old question of Brady and produces many finitely presented non-hyperbolic subgroups of hyperbolic groups. This is joint work with Pierre Py. |
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Thursday October 20, 14:05 (UK time), B3.02 Zeeman. Henry Bradford (Cambridge) Local permutation stability |
Abstract: A group \(\Gamma\) is sofic if elements of \(\Gamma\) can be distinguished by almost-actions on finite sets. It is a major unsolved problem to determine whether all groups are sofic. One approach to this problem which has gained much recent attention is that of “permutation stability”, that is, showing that almost-actions of a group are controlled by its actions. We introduce a “local” generalization of permutation stability, under which actions are replaced by partial actions. We exhibit an uncountable family of groups which are locally permutation stable but not permutation stable, coming from topological dynamics. The proof is based on a criterion for local stability of amenable groups, in terms of invariant random subgroups. |
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Thursday October 27, 14:05 (UK time), B3.02 Zeeman. Daniel Berlyne (Bristol) Braid groups of graphs |
Abstract: 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 right-angled Artin group. |
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Thursday November 3, 14:05 (UK time), B3.02 Zeeman. Becca Winarski (College of the Holy Cross) Polynomials, branched covers, and trees |
Abstract: Thurston proved that a post-critically 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. When it is, we determine which polynomial it is equivalent to. This is joint work with Jim Belk, Justin Lanier, and Dan Margalit. |
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Thursday November 10, 14:05 (UK time), B3.02 Zeeman. None (None) None |
Abstract: None |
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Thursday November 17, 14:05 (UK time), B3.02 Zeeman. Bradley Zykoski (University of Michigan) 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 $\GL(2,\RR)$-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 Eskin-Mirzakhani-Mohammadi and the Cylinder Deformation theorem of Wright. Investigation of the topology of strata is still in its nascency, although recent work of Calderon-Salter and Costantini-Möller-Zachhuber indicate that this field is rapidly blossoming. In this talk, I will discuss a way of decomposing strata into finitely many higher-dimensional 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 $\GL(2,\RR)$-action. |
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Thursday November 24, 14:05 (UK time), B3.02 Zeeman. None (None) None |
Abstract: None |
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Thursday December 1, 14:05 (UK time), B3.02 Zeeman. Koji Fujiwara (Kyoto) Growth rates 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 well-ordered, and that each number can be the rate of growth of at most finitely many generating sets up to automorphism of the group. If there is time, I may also discuss generalisation to acylindrically hyperbolic groups. |
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Thursday December 8, 14:05 (UK time), B3.02 Zeeman. Ric Wade (Oxford) \(\Aut\)-invariant quasimorphisms on groups |
Abstract: For a large class of groups, we exhibit an infinite-dimensional space of homogeneous quasimorphisms that are invariant under the action of the automorphism group. This class includes non-elementary hyperbolic groups, infinitely-ended finitely generated groups, some relatively hyperbolic groups, and a class of graph products of groups that includes all right-angled Artin and Coxeter groups that are not virtually abelian. This is joint work with Francesco Fournier-Facio. |