Spring 2023
Meeting dates: Tuesdays (Topology Seminar) 1:50-2:50 pm at Dulles Hall 016 and
Thursdays (Geometric Group Theory Seminar) from 1:50 to 2:50 pm at Caldwell Laboratory 0115
Organizers - Arman Darbinyan, Jingyin Huang, Heejoung Kim, Christoforos Neofytidis, and Francis Wagner
The seminars will be run partially in person and partially virtually over Zoom for the Spring 2023 Semester
https://osu.zoom.us/j/94449451323?pwd=RXZ5am9hYzVlaHVEaGNNRnpMbkRsQT09
Meeting ID: 944 4945 1323 Password: 779408
January 10- Geunho Lim (UC Santa Barbara)G
Bounds on rho-invariants and simplicial complexity of triangulated manifolds
In this talk, we show the existence of linear bounds on various rho-invariants. In particular, we construct a desired cobordism over a group, whose complexity is linearly bounded by that of its boundary. Employing a combinatorial concept of G-colored polyhedra, we give linear bounds on Atiyah-Singer invariants of PL manifolds. Using relative hyperbolization, we obtain linear bounds on Cheeger-Gromov invariants of PL manifolds endowed with a faithful representation. As applications, we give concrete examples in the complexity theory of high-dimensional (homotopy) lens spaces. This is a joint work with Shmuel Weinberger.
January 17- Arman Darbinyan
Geometric interpretation of Turing degrees and some applications in geometric group theory
Classical theorems of Novikov, Boone, and Higman establish important connections between recursive and recursively enumerable sets on the one hand and the theory of finitely presented groups on the other hand. Despite the groundbreaking nature of these results, not much is known about possible extensions of those results to the realm of higher Turing degrees. My talk will be a contribution in this direction.
In my talk, I will discuss how one can view Turing degrees through invariants coming from geometric group theory. In particular, I will introduce a new quasi-isometric invariant and will show how this invariant can grasp Turing degrees of arbitrary degrees. Through these means, we will also introduce an interesting special class of functions, called Turing saturated maps, which will be useful for our applications.
As a main application, we will discuss a purely geometric group theoretical result that establishes quasi-isometric diversity of f.g. left-orderable simple groups, which is an important class of groups discovered by Hyde and Lodha. Previous attempts to obtain this result through more usual geometric/algebraic tools have not succeeded, which emphasizes the importance of the idea of Turing degrees in this context.
January 19- Feng Zhu (UW Madison)
Relatively Anosov representations
Putting hyperbolic metrics on a finite-type surface S gives us representations of the fundamental group of S into PSL(2,R) with many nice geometric and dynamical properties: for instance they are discrete and faithful, and in fact stably quasi-isometrically embedded.
In this talk, we will introduce (relatively) Anosov representations, which generalise this picture to higher-rank Lie groups such as PSL(d,R) for d>2, giving us a class of (relatively) hyperbolic subgroups there with similarly good geometric and dynamical properties.
February 2-Bradley Zykoski (University of Michigan)
A polytopal decomposition of strata of translation surfaces
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 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,R)-action.
February 23- Piotr Przytycki (McGill) (Talk cancelled)
Coxeter groups are biautomatic.
Coxeter groups are groups generated by involutions si, with the relations of form (sisj)m=id. For each Coxeter group, we will be discussing a particular system of "voracious" paths between any two vertices of the Cayley graph. This system turns out to have the "fellow traveller" property and is generated by a finite state automaton. This is joint work with Damian Osajda.
March 2- Samuel Corson (UPV/EHU )
Many trivially generated groups are the same
When the algebraic generators of a group are all trivial then the group is also trivial. However, when the group is combinatorially described using infinite words, the "generators" can be trivial while the group is uncountable. I will give some background and present recent work showing that many natural examples of such groups are isomorphic.
March 7- Francis Wagner
Quasilinear Estimates of S-machine Complexity
An S-machine is a computational model which resembles a multi-tape, non-deterministic Turing machine that works over group words rather than the "positive words" comprising the corresponding free monoid. This model has proved fruitful since its introduction in 2002, playing a key role in the solution of several long-standing open problems in group theory. We will present an equivalent graphical formulation of S-machines which produces objects that are both easily understood and readily `combined', allowing for better estimates on the object's computational complexity. Finally, we will discuss some implications of this, including a fundamental link between word problem decidability and Dehn function.
April 11- Matt Zaremsky (SUNY, Albany)
Shift-similar groups of permutations of the natural numbers
In joint work with Brendan Mallery (Tufts), we introduce the notion of a "shift-similar" subgroup of the group of permutations of the natural numbers N. The definition makes use of the fact that any cofinite subset of N is canonically bijective with N, and is an analog to the well-known condition of "self-similarity" for subgroups of the group of automorphisms of a tree. In this talk I will discuss self-similarity and shift-similarity, compare and contrast them, and mention connections to the world of Thompson groups and Houghton groups. Perhaps the most striking result is that there exist uncountably many isomorphism classes of finitely generated shift-similar groups, unlike the self-similar case.
April 13-Kevin Schreve (LSU)
Homology growth, hyperbolization, and fibering.
I will talk about joint work with Grigori Avramidi and Boris Okun where we construct closed, aspherical 7-manifolds with word hyperbolic, special fundamental group that do not virtually fiber over the circle. The obstruction to fibering is linear F_p-homology growth in residual.towers of regular finite covers for an odd prime p. In previous work, we constructed locally CAT(0) examples, and I will describe what needs to be done to “hyperbolize” the construction.
April 18-Pallavi Dani (LSU)
Non-quasiconvex subgroups of hyperbolic groups
I will talk about joint work with Ivan Levcovitz, in which we provide a new method of constructing non-quasiconvex subgroups of hyperbolic groups. The hyperbolic groups constructed are in the natural class of right-angled Coxeter groups, and can be chosen to be 2-dimensional. We use techniques inspired by Stallings folds to establish the non-quasiconvexity of our subgroups.
April 20- Caleb Dilsavor (OSU)
Thermodynamic formalism for non-compact locally CAT(-1) spaces
I will discuss an upcoming joint work with Daniel Thompson in which we extend the Patterson-Sullivan construction of equilibrium states to geodesic flows of non-compact locally CAT(-1) spaces. The result we will generalize, which was proved by Paulin Pollicott Schapira in the Riemannian setting, states that a Gibbs measure always exists, it is the unique equilibrium state if it is finite, and if it is not finite then there are no equilibrium states. Paulin Policott Schapira remarked that even extending the construction of the Gibbs measure to the CAT(-1) setting seemed difficult due to discrepancies between different extensions of geodesic segments. In our paper we will show that these discrepancies only introduce bounded errors which make the CAT(-1) space ‘behave’ as if it were Gromov hyperbolic, and that nonetheless one can employ Coornaert’s Patterson-Sullivan theory to construct a Gibbs measure. This Gibbs measure then can be shown to satisfy the Paulin Pollicott Schapira result after some care is taken with the bounded errors.
In my talk I will focus on the construction of the invariant measure, in a generalized form for Gromov hyperbolic spaces. In particular I will explain what the analogue of a potential function from thermodynamics should be for a hyperbolic group, and what this looks like on the Gromov-Mineyev flow.