The seminars will be run virtually over Zoom through the Autumn 2020 Semester
https://osu.zoom.us/j/93661626526?pwd=b2xiSEJSTm9BRVlRSitOZXVkMVMzZz09
Zoom ID: 936 6162 6526, Password: 273789
August 25-
Sam Shepherd (University of Oxford)
Quasi-isometric rigidity of generic cyclic HNN extensions of free groups
Studying quasi-isometries between groups is a major theme in geometric group theory. Of particular interest are the situations where the existence of a quasi-isometry between two groups implies that the groups are equivalent in a stronger algebraic sense, such as being commensurable. I will survey some results of this type, and then talk about recent work with Daniel Woodhouse where we prove quasi-isometric rigidity for certain graphs of virtually free groups, which include "generic" cyclic HNN extensions of free groups.
August 27-
Waltraud Lederle (Université Catholique de Louvain)
Conjugacy in Neretin's group
We explain when two almost automorphisms of a regular tree are conjugate. Our main focus will be on non-elliptic elements, where we can use strand diagrams introduced by Belk and Matucci to describe conjugacy in Thompson's V. This is joint work with Gil Goffer.
September 1-
Weiyi Zhang (University of Warwick)
From smooth to almost complex
An almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. We will discuss differential topology of almost complex manifolds, explain how to use transversality statements for smooth manifolds to formulate and prove corresponding results for an arbitrary almost complex manifold. The examples include intersection of almost complex manifolds, structure of pseudoholomorphic maps and zero locus of certain harmonic forms.
One of the main technical tools is Taubes’ notion of “positive cohomology assignment”, which plays the role of local intersection number. I will begin with explaining its motivation through multiplicities of zeros of a smooth function.
Our results would lead to a notion of birational morphism between almost complex manifolds. Various birational invariants, including Kodaira dimension, for almost complex manifolds will be introduced and discussed (this part is joint with Haojie Chen).
September 3-
September 8-
September 10-
Mark Pengitore (OSU)
Coarse embeddings and homological filling functions
We will demonstrate that a coarse embedding of a group into a group of geometric dimension 2 induces an inequality on homological Dehn functions in dimension 2. As an application of this result, we are able to show that if a finitely presented group coarsely embeds into a hyperbolic group of geometric dimension 2, then it is hyperbolic. Another application is a characterization of subgroups of group geometric dimension with quadratic Dehn fucntion. If there is enough time, we will talk about various higher dimensional generalizations of the main result. This is all joint work with Rob Kropholler.
September 15-
Michele Chu (UIC)
Virtual torsion in the homology of 3-manifolds
Hongbin Sun showed that a closed hyperbolic 3-manifold virtually contains any prescribed torsion subgroup as a direct factor in homology. In this talk we will discuss joint work with Daniel Groves generalizing Sun’s result to irreducible 3-manifolds which are not graph-manifolds.
September 17-
September 22-
Lvzhou Chen (University of Texas Austin)
Stable commutator lengths of integral chains in right-angled Artin groups
It follows from theorems of Agol and Kahn-Markovic that the fundamental group of any closed hyperbolic 3-manifold contains a special subgroup of finite index. Very little is known about how large the index needs to be. Motivated by this, in this joint work with Nicolaus Heuer, we study stable commutator lengths (scl) of integral chains in right-angled Artin groups (RAAGs). Topologically, an integral 1-chain in a group G is a collection of loops in the K(G,1) space with integral weights, and its scl is the least complexity of surfaces bounding the weighted loops. We show that the infimal positive scl of integral chains in any RAAG is positive, and its size explicitly depends on the defining graph of the RAAG up to a multiplicative constant 12. In particular, the size is non-uniform among RAAGs, which is unexpected.
September 24-
September 29-
Benjamin Steinberg (City College of New York)
Simplicity of Nekrashevych algebras of contracting self-similar groups
A self-similar group is a group $G$ acting on the Cayley graph of a finitely generated free monoid $X^*$ (i.e., regular rooted tree) by automorphisms in such a way that the self-similarity of the tree is reflected in the group. The most common examples are generated by the states of a finite automaton. Many famous groups like Grigorchuk's 2-group of intermediate growth are of this form.
Nekrashevych associated $C^*$-algebras and algebras with coefficients in a field to self-similar groups. In the case $G$ is trivial, the algebra is the classical Leavitt algebra, a famous finitely presented simple algebra.
Nekrashevych showed the algebra associated to the Grigorchuk group is not simple in characteristic 2, but Clark, Exel, Pardo, Sims and Starling showed its Nekrashevych algebra is simple over all other fields. Nekrashevych then showed that the algebra associated to the Grigorchuk-Erschler group is not simple over any field (the first such example).
The Grigorchuk and Grigorchuk-Erschler groups are contracting self-similar groups. This important class of self-similar groups includes Gupta-Sidki p-groups and many iterated monodromy groups like the Basilica group. Nekrashevych proved algebras associated to contacting groups are finitely presented.
In this talk we discuss a recent result of the speaker and N. Szakacs (York/Szeged) characterizing simplicity of Nekrashevych algebras of contracting groups. In particular, we give an algorithm for deciding simplicity given an automaton generating the group. We apply our results to several families of contracting groups like Gupta-Sidki groups and Sunic's generalizations of Grigorchuk's group associated to polynomials over finite fields.
October 1-
Priyam Patel (University of Utah)
Isometry groups of infinite-genus hyperbolic surfaces
Allcock, building on the work of Greenburg, proved that for any countable group G, there is a a complete hyperbolic surface whose isometry group is exactly G. When the group is finite, Allcock’s construction yields a closed surface, but when it is not finite, the construction gives an infinite-genus surface.
In this talk, we discuss a related question. We fix any infinite-genus surface S and characterize all groups that can arise as the isometry group for a complete hyperbolic structure on S. If time allows, we give two applications of the main result. This talk is based on joint work with T. Aougab and N. Vlamis.
October 6-
Emily Stark (Wesleyan University)
Deep homology and obstructing group actions
Studying the topology of a space "at infinity" offers a powerful perspective in geometric group theory. To capture the structure at infinity relative to a subspace, one can consider relative ends and associated deep homology groups. A goal is then to compute these homology groups and use homological arguments to study these pairs of spaces. In this talk, I will explain how both goals are possible in the setting of coarse embeddings into coarse PD(n) spaces. As applications, we use homological arguments to prove that certain groups cannot act properly on a given manifold. This is joint work with Chris Hruska and Hung Cong Tran.
October 8-
October 13-
Matt Cordes (ETH)
Geometric approximate group theory
An approximate group is a group that is "almost closed" under multiplication. Finite approximate subgroups play a major role in additive combinatorics. Recently Breuillard, Green and Tao have established a structure theorem concerning finite approximate subgroups and used this theory to reprove Gromov’s growth theorem. Infinite approximate groups were studied implicitly long before the formal definition. Approximate subgroups of R^n that are Delone sets can be constructed using "cut-and-project" methods and are models for mathematical quasi-crystals. Recently, Björklund and Hartnick have begun a program investigating infinite approximate lattices in locally compact second countable groups using geometric and measurable structures. In the talk I will introduce infinite approximate groups and their geometric aspects. This is joint work with Hartnick and Tonic.
October 15-
Laurent Bartholdi (University of Goettingen)
Domino problems on graphs and groups
For a fixed edge-labelled graph, the "domino problem" asks: "given a collection of labelled dominoes (with numbers on their ends), can one put a domino on each edge of the graph in such a manner that edge labels and vertex numbers match?''
In spite of its naive appearence, this problem is deeply connected to (monadic, second-order) logic; remarkably, it is undecidable for graphs such as the infinite square grid – the "Wang tiling problem".
I will consider it on graphs produced from a group action: Cayley graphs, Schreier graphs. I will exhibit a class of graphs for which the problem is decidable, as well as interesting examples not containing grids yet also having undecidable domino problem.
Part of this is joint work with Ville Salo.
October 20-
October 22-
Josiah Oh (OSU)
Geometry of non-transitive graphs
Due to Freudenthal and Hopf it is known that finitely generated groups have zero, one, two, or infinitely many ends, and that a finitely generated group has two ends if and only if it is virtually infinite cyclic. Analogous statements have since been proved for the more general class of vertex-transitive graphs (graphs whose automorphism groups act transitively on their vertices). In this talk we introduce a notion of coarse transitivity for graphs and show that statements analogous to the Freudenthal-Hopf theorems also hold for coarsely transitive graphs.
Trofimov proved that a vertex-transitive graph of polynomial growth is necessarily quasi-isometric to a finitely generated (nilpotent) group. In contrast, we show that for each finitely generated group $G$, there exist continuum many pairwise non-quasi-isometric regular graphs that have the same growth rate, number of ends, and asymptotic dimension as $G$. This is joint work with Mark Pengitore.
October 27
Mauricio Bustamante (University of Cambridge)
Diffeomorphisms of solid tori
The homotopy groups of the diffeomorphism group of a high dimensional manifold with infinite fundamental group can be infinitely generated. The simplest example of this sort is the solid torus T=S^1\times D^{d-1}. In fact, using surgery and pseudoisotopy theory, it is possible to show that in the range of degrees up to (roughly) d/3, the homotopy groups of Diff(T) contain infinitely generated torsion subgroups.
In this talk, I will discuss an alternative point of view to study Diff(T) which does not invoke pseudoisotopy theory: when d=2n, we interpret Diff(T) as the "difference" between diffeomorphisms and certain self-embeddings of the manifold X_g obtained as the connected sum of T with the g-fold connected sum of S^n \times S^n.
We will see how infinitely generated torsion subgroups appear from this perspective, and that they can be found even up to degrees d/2. This is ongoing joint work with O. Randal-Williams.
October 29-
Alex Furman (UIC)
Looking at negatively curved manifolds from the outside.
Given a negatively curved closed manifold $(M,g)$ one can look at the length function $L_{g,x}(\gamma)$ on the fundamental group $\Gamma=\pi_1(M)$ given by the displacement $L_{g,x}(\gamma)=dist(\gamma.x, x)$ on the universal cover $\tilde{M}$ of $M$. Identifying length functions that differ by a bounded amount, we erase the dependence on $x$. It turns out that a lot (conjecturally) all of the geometry of $(M,g)$ can be read from the associated (class of) the length function. Moreover, the setting can be vastly extended to include a variety of length functions on Gromov hyperbolic spaces.
In the talk, I will describe some results and discuss some interesting problems pertaining to the general setting and to concrete examples such as surface groups.
November 3-
Jonas Dere (KU Leuven Kulak)
Strongly scale-invariant virtually polycyclic groups
A finitely generated group is called strongly scale-invariant if there exists an injective endomorphism on the group with image of finite index and such that the intersection of the images of the iterations is trivial. The only known examples of such groups are virtually nilpotent, or equivalently, all examples have polynomial growth. A question by Nekrashevych and Pete asks whether these groups are the only possibilities for such endomorphisms, motivated by the positive answer due to Gromov in the special case of expanding group morphisms. In this talk, we will discuss this question in the special case of virtually polycyclic groups, by using the algebraic hull of these groups.
November 5-
Marcin Sabok (McGill University)
Hyperfiniteness at Gromov boundaries
I will discuss recent results establishing hyperfiniteness of equivalence relations induced by actions on Gromov boundaries of various hyperbolic spaces. This includes boundary actions of hyperbolic groups (joint work with T. Marquis) and actions of the mapping class group on boundaries of the arc graph and the curve graph (joint work with P. Przytycki)
November 10-
Alex Margolis (Vanderbilt University)
Topological completions of quasi-actions and discretisable spaces
A fundamental problem in geometric group theory is the study of quasi-actions. We introduce and investigate discretisable spaces: spaces for which every cobounded quasi-action can be quasi-conjugated to an isometric action on a locally finite graph. Work of Mosher-Sageev-Whyte shows that non-abelian free groups are discretisable, but the property holds much more generally. For instance, every non-elementary hyperbolic group that is not virtually isomorphic to a cocompact lattice in rank one Lie group is discretisable.
Along the way, we study the coarse geometry of groups containing almost normal/commensurated subgroups, and we introduce the concept of the topological completion of a quasi-action. The topological completion is a locally compact group, well-defined up to a compact normal subgroup, reflecting the geometry of the quasi-action. We give several applications of the tools we develop. For instance we show that any finitely generated group quasi-isometric to a Z-by-hyperbolic group is also Z-by-hyperbolic, and prove quasi-isometric rigidity for a large class of right-angled Artin groups.
November 12-
Alex Rasmussen (University of Utah)
Actions of certain solvable groups on hyperbolic metric spaces
In this talk I will discuss the classification of the cobounded hyperbolic actions of solvable Baumslag-Solitar groups and lamplighter groups. These groups have actions on trees which have convenient descriptions using ring theory. I will then discuss current work on generalizing this ring theoretic machinery, and possible applications to certain abelian-by-cyclic groups. This is joint work with Carolyn Abbott and Sahana Balasubramanya.
November 17-
Xiaolei Wu (Bielefeld University)
On the poly-freeness of Artin groups
Artin group is an important class of groups under intensive study in recent years. It is a generalization of the braid groups. Bestvina asks whether all Artin groups are virtually poly-free. In this talk, we first give an introduction to poly-free groups and Artin groups. We explain some connections with the Farrell-Jones Conjecture. Then we explain some recent progress of Bestvina's question. In particular, we will give a short proof of the fact that Even Artin groups of FC-type are polyfree. Part of this is joint work with Benjamin Brück and Dawid Kielak.
November 19-
Daniel Woodhouse (University of Oxford)
Action rigidity of free products of hyperbolic manifold groups
Gromov's program for understanding finitely generated groups up to their large scale geometry considers three possible relations: quasi-isometry, abstract commensurability, and acting geometrically on the same proper geodesic metric space. A *common model geometry* for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is *action rigid* if any group G' that has a common model geometry with G is abstractly commensurable to G. We show that free products of closed hyperbolic surface or 3-manifold groups are action rigid. As a corollary, we obtain torsion-free, Gromov hyperbolic groups that are quasi-isometric, but do not even virtually act on the same proper geodesic metric space.
December 1-
Jason Manning (Cornell University)
Perturbing the action of a hyperbolic group on its boundary
A hyperbolic group G comes with an action by homeomorphisms on its Gromov boundary. In general this boundary is some compact metrizable space which can have complicated local topology, but sometimes it is a sphere (for example ifG is the fundamental group of a closed negatively curved manifold). We show that the action of a torsion free hyperbolic group on its boundary is topologically stable, assuming that boundary is a sphere. This generalizes work of Bowden--Mann in the pinched negative curvature case, and is joint work with Kathryn Mann.
December 3-
Ilir Snopce (Federal University of Rio de Janeiro)
Retracts in free groups
A subgroup R of a group G is said to be a retract of G if there is a homomorphism r : G → R that restricts to the identity on R. I will talk about retracts in free groups. In particular, I will discuss the following question raised by Bergman: Let F be a free group of finite rank and let R be a retract of F. Is it H ∩ R is a retract of H for every finitely generated subgroup H of F?
This talk is based on a joint work with Slobodan Tanushevski and Pavel Zalesskii.