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Geometry and Topology

Please contact Saul Schleimer or Robert Kropholler if you would like to speak or to suggest a speaker.A cubical cat

We will also attempt to maintain an up-to-date listing at

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. All of the talks will be streamed at this link.

    Click on a title to view the abstract!

    Upcoming Seminars

  • 29 February 2024 at 14:00 in B3.02

    Speaker: Joe MacManus (University of Oxford)

    Title: Groups quasi-isometric to planar graphs

    Abstract: A classic and important theorem originating in work of Mess states that a f.g. group is quasi-isometric to a complete Riemannian plane if and only if it is a virtual surface group. Another related result obtained by Maillot states that a f.g. group is virtually free if and only if it is quasi-isometric to a complete planar simply connected Riemannian surface with non-compact geodesic boundary. These results illustrate the general philosophy that planarity is a very `rigid' property amongst f.g. groups.

    In this talk I will build on the above and sketch how to characterise those f.g. groups which are quasi-isometric to planar graphs. Such groups are virtually free products of free and surface groups, and thus virtually admit a planar Cayley graph. The main technical step is proving that such a group is accessible, in the sense of Dunwoody and Wall. This is achieved through a careful study of the dynamics of quasi-actions on planar graphs.

  • 07 March 2024 at 14:00 in B3.02

    Speaker: Marco Linton (University of Oxford)

    Title: TBA

    Abstract: TBA

  • 14 March 2024 at 14:00 in B3.02

    Speaker: Davide Spriano (University of Oxford)

    Title: TBA

    Abstract: TBA

  • 09 May 2024 at 14:00 in B3.02

    Speaker: Rohini Ramadas (University of Warwick)

    Title: TBA

    Abstract: TBA

  • Past Seminars

  • 01 February 2024 at 14:00 in B3.02

    Speaker: Samuel Shepherd (Vanderbilt University)

    Title: One-ended halfspaces in group splittings

    Abstract: I will introduce the notion of halfspaces in group splittings and discuss the problem of when these halfspaces are one-ended. I will also discuss connections to JSJ splittings of groups, and to determining whether groups are simply connected at infinity. This is joint work with Michael Mihalik.

  • 25 January 2024 at 14:00 in B3.02

    Speaker: Francesco Fournier-Facio (University of Cambridge)

    Title: Infinite simple characteristic quotients

    Abstract: The rank of a finitely generated group is the minimal size of a generating set. Several questions that received a lot of attention around 50 years ago ask about the rank of finitely generated groups, and how this relates to the rank of their direct powers. In this context, Wiegold asked about the existence of infinite simple characteristic quotients of free groups. I will review this framework, present several open questions – old and new – and present a solution to Wiegold’s problem.

    Joint with Rémi Coulon

  • 18 January 2024 at 14:00 in B3.02

    Speaker: Ian Leary (University of Southampton)

    Title: Residual finiteness of generalized Bestvina-Brady groups

    Abstract: (joint with Vladimir Vankov)
    I discovered/created generalized Bestvina-Brady groups to give an uncountable family
    of groups with surprising homological properties. In this talk, I will introduce the
    groups and describe joint work with Vladimir Vankov addressing the following questions:
    when are they virtually torsion-free?
    when are they residually finite?
    This leads naturally to a third question:
    when do they virtually embed in right-angled Artin groups?
    There are nice conjectural answers to all three questions, which we have proved in
    some cases.

  • 11 January 2024 at 14:00 in B3.02

    Speaker: Richard Wade (University of Oxford)

    Title: Quasi-flats in the Aut free factor complex

    Abstract: We will describe families of quasi-flats in the "$Aut(F_n)$ version" of the free factor complex. This shows that, unlike its more popular "Outer" cousin, the Aut free factor complex is not hyperbolic. The flats are reasonably simple to describe and are shown to be q.i. embedded via the construction of a coarse Lipschitz retraction. This leaves many open problems about the coarse geometry of this space, and I hope to talk about a few of them. This is joint work with Mladen Bestvina and Martin Bridson.

  • 07 December 2023 at 14:00 in B3.02

    Speaker: Sam Hughes (University of Oxford)

    Title: Centralisers and classifying spaces for Out(F_N)

    Abstract: In this talk I will outline reduction theory for mapping classes and explain various attempts to construct similar machinery for elements of Out(F_N). I will then present a new reduction theory for studying centralisers of elements in IA_3(N), the finite index level 3 congruence subgroup of Out(F_N). Using this I will explain an application to the classifying space for virtually cyclic subgroups, a space notable for its appearance in the Farrell--Jones Conjecture. Based on joint work with Yassine Guerch and Luis Jorge Sánchez Saldaña.

  • 30 November 2023 at 14:00 in B3.02

    Speaker: Cameron Rudd (MPIM Bonn)

    Title: Stretch laminations and hyperbolic Dehn surgery

    Abstract: Given a hyperbolic manifold M and a homotopy class of maps from M to the circle, there is an associated geodesic "stretch" lamination encoding at which points in M the Lipschitz constant of any map in the homotopy class must be large. Recently, Farre-Landesberg-Minsky related these laminations to horocycle orbit closures in infinite cyclic covers and when M is a surface, they analyzed the possible structure of these laminations. I will discuss the case where M is a 3-manifold and give the first 3-dimensional examples where these laminations can be identified. The argument uses the Thurston norm and tools from quantitative Dehn surgery.

  • 23 November 2023 at 14:00 in B3.02

    Speaker: Jeffrey Giansiracusa (University of Durham)

    Title: Topology of the matroid Grassmannian

    Abstract: The matroid Grassmannian is the moduli space of oriented matroids; this is an important combinatorial analogue of the ordinary oriented real Grassmannian. Thirty years ago MacPherson showed us that understanding the homotopy type of this space can have significant implications in manifold topology, such as providing combinatorial formulae for the Pontrjagin classes. In some easy cases, the matroid Grassmannian is homotopy equivalent to the oriented real Grassmannian, but in most cases we have no idea whether or not they are equivalent. This question is known as MacPherson's conjecture. I'll show that one of the important homotopical structures of the oriented Grassmannians has an analogue on the matroid Grassmannian: the direct sum monoidal product (which gives rise to topological K-theory) is E-infinity.

  • 16 November 2023 at 14:00 in B3.02

    Speaker: Rob Kropholler (Warwick)

    Title: The landscape of Dehn functions

    Abstract: -

  • 09 November 2023 at 14:00 in B3.02

    Speaker: Monika Kudlinska (University of Oxford)

    Title: Subgroup separability in 3-manifold and free-by-cyclic groups

    Abstract: A group G is said to be subgroup separable if every finitely generated subgroup of G is the intersection of finite index subgroups. It is known that a fundamental group of a compact, irreducible, closed 3-manifold M is subgroup separable if and only if M is geometric. We will discuss the problem of subgroup separability in free-by-cyclic groups by drawing a parallel between free-by-cyclic and 3-manifold groups. Time permitting, we will discuss how to extend these ideas to find non-separable subgroups in random groups

  • 02 November 2023 at 14:00 in B3.02

    Speaker: Adele Jackson (University of Oxford)

    Title: Algorithms for Seifert fibered spaces

    Abstract: Given two mathematical objects, the most basic question is whether they are the same. We will discuss this question for triangulations of three-manifolds. In practice there is fast software to answer this question and theoretically the problem is known to be decidable. However, our understanding is limited and known theoretical algorithms could have extremely long run-times. I will describe a programme to show that the 3-manifold homeomorphism problem is in the complexity class NP, and discuss the important sub-case of Seifert fibered spaces.

  • 19 October 2023 at 14:00 in B3.02

    Speaker: Clément Legrand (LaBRI)

    Title: Reconfiguration of square-tiled surfaces

    Abstract: A square-tiled surface is a special case of a quadrangulation of a surface, that can be encoded as a pair of permutations in \(S_n \times S_n\) that generates a transitive subgroup of \(S_n\). Square-tiled surfaces can be classified into different strata according to the total angles around their conical singularities. Among other parameters, strata fix the genus and the size of the quadrangulation. Generating a random square-tiled surface in a fixed stratum is a widely open question. We propose a Markov chain approach using "shearing moves": a natural reconfiguration operation preserving the stratum of a square-tiled surface. In a subset of strata, we prove that this Markov chain is irreducible and has diameter \(O(n^2)\), where \(n\) is the number of squares in the quadrangulation.

  • 12 October 2023 at 14:00 in B3.02

    Speaker: Mark Pengitore (University of Virginia)

    Title: Residual finiteness growth functions of surface groups with respect to characteristic quotients

    Abstract: Residual finiteness growth functions of groups have attracted much interest in recent years. These are functions that roughly measure the complexity of the finite quotients needed to separate particular group elements from the identity in terms of word length. In this talk, we study the growth rate of these functions adapted to finite characteristic quotients. One potential application of this result is towards linearity of the mapping class group

  • 05 October 2023 at 14:00 in B3.02

    Speaker: Raphael Zentner (Durham University)

    Title: Rational homology ribbon cobordism is a partial order

    Abstract: Last year, Ian Agol has proved that ribbon knot concordance is a partial order on knots, a conjecture that has been open for more than three decades. His proof is beautiful and surprisingly simple. There is an analog notion of ribbon cobordism for closed 3-manifolds. We use Agol's method to show that this notion of ribbon cobordism is also a partial order within the class of irreducible 3-manifolds. This is joint work with Stefan Friedl and Filip Misev.