McGill Geometric Group Theory Seminar
The seminar takes place on Wednesday at 3 PM in 920 Burnside Hall, unless otherwise noted below.
Seminar organizers: William Chong, Zachary Munro, Daniel Wise
Seminar organizers: William Chong, Zachary Munro, Daniel Wise
Button observed that finitely generated linear groups containing no nontrivial unipotent matrices behave much like groups admitting proper actions by semisimple isometries on complete CAT(0) spaces. It turns out that any finitely generated linear group possesses an action on such a space whose restrictions to unipotent-free subgroups are in some sense tame. We discuss this phenomenon and some of its implications for the representation theory of certain 3-manifold groups.
We will give definitions and results in the theory of hyperbolic groups with the purpose of highlighting some central themes of geometric group theory. Topics will certainly include: the definition of a hyperbolic group, geometric action, quasi-isometry, geodesic stability, boundary, etc.
We will introduce the small cancellation theory, which is a convenient tool to generate examples. We first examine the essence of the proof about the word problem for the fundamental group of the orientable surface of genus 2. We then define a metric small cancellation condition and sketch a proof for the associated word problems with disk diagrams. Finally, we state other related small cancellation conditions and explore some of their applications.
This will be the first of 2 talks. In this talk I will just introduce some of the notions behind much of my research in geometric group theory.
In the talk, we will introduce a family of finitely generated residually finite groups. These groups are doubles of a rank-2 free group along an infinitely generated subgroup H. Varying H yields uncountably many groups up to isomorphism. This is a joint work with Daniel Wise.
Dani will tell us everything he knows about residual finiteness in one hour. The seminar will be introductory in nature, so those new to GGT are especially invited to participate.
Given a finite-sheeted, possibly branched covering space between surfaces, it’s natural to ask how the mapping class group of the covering surface relates to the mapping class group of the base surface. In this talk, we will take a journey through this question for surfaces with boundary. It will feature appearances from the fundamental groupoid, the Birman-Hilden theorem, the Burau representation and new embeddings of the braid group in mapping class groups.
An error-correcting code is locally testable (LTC) if there is a random tester that reads only a constant number of bits of a given word and decides whether the word is in the code, or at least close to it.
A long-standing problem asks if there exists such a code that also satisfies the golden standards of coding theory: constant rate and constant distance. Unlike the classical situation in coding theory, random codes are not LTC, so this problem is a challenge of a new kind.
We construct such codes based on what we call (Ramanujan) Left/Right Cayley square complexes. These 2-dimensional objects seem to be of independent interest.
The lecture will be self-contained.
In joint work with Y. Algom-Kfir, D. Gagnier, I. Kapovich, J. Maher, S. J. Taylor, we look from various perspectives at what a typical element of the outer automorphism group of a free group (Out(F_r)) "looks like." This builds on observed connections between how Out(F_r) conjugacy class invariants arise via iterating loops in graphs & how this impacts trajectories of geodesics in Culler-Vogtman Outer space. Having attended Part 1 of this talk will be useful, but there will be plenty of images to gaze out (& likely found beautiful) even if one hadn't attended it.
I will discuss the dictionary between the algebraic structure of a right-angled Artin group and the combinatorics of the defining graph. I will then use the cohomology of a right-angled Artin group to provide a characterization of Hamiltonicity of the underlying graph.
Dani will again tell us everything he knows about residual finiteness in one hour. The seminar will be introductory in nature, so those new to GGT are especially invited to participate.
Roughly speaking, a subgroup of a lattice in a semisimple Lie group is said to be thin if the subgroup is of infinite index in the lattice but is Zariski-dense in the Lie group. Free groups and surface groups have many manifestations as thin subgroups of lattices in Lie groups, by classical work of Tits in the free case, and by work of Kahn–Markovic, Hamenstädt, Long–Reid, Kahn–Labourie–Mozes, and others in the surface group case. We sketch an argument that an irreducible right-angled Coxeter group on n>2 vertices embeds as a thin subgroup of an arithmetic lattice in O(p,q) for some p,q>0 satisfying p+q=n, and that we can arrange for the lattice to be cocompact.
Dani will again tell us everything he knows about residual finiteness in one hour, and he will end the series here. The seminar will be introductory in nature, so those new to GGT are especially invited to participate.
Howie (2002) proved that any one-relator product of 3 cyclic groups is never a trivial group, via constructing a non-trivial representation to $SO(3)$. After stating the conjecture, we first focus on $S^1$-equivariant homotopy. Then, we describe a sketch of the proof. We may talk about some applications or generalizations if time allows.
Reference: Howie, J. (2002). A proof of the scott–wiegold conjecture on free products of cyclic groups. Journal of Pure and Applied Algebra, 173(2), 167–176. https://doi.org/10.1016/s0022-4049(02)00042-7
Dani will tell us interesting definitions, main theorems and open questions in the theory of one-relator groups. The seminar will be introductory in nature, so those new to GGT are especially invited to participate.
Among other finiteness properties, Hruska-Wise showed that Sageev's construction of the dual cube complex allows one to construct relatively cocompact actions of relatively hyperbolic groups on CAT(0) cube complexes. In this talk, we describe Sageev's construction and prove relative compactness for relatively hyperbolic groups.
For a group G, the generalized conjugacy problem asks if there exists an algorithm that takes as input two finite ordered lists $(a_1,...,a_n), (b_1,...,b_n)$ of group elements of the same size and determines if there exists a group element simultaneously conjugating each $a_i$ to $b_i$ in $G$. In this talk, we survey the work of Bridson and Howie in https://people.maths.ox.ac.uk/bridson/papers/BHowIJAC/BHijac.pdf (and we may also touch on the ideas of Buckley and Holt in https://arxiv.org/pdf/1111.1554.pdf, if time permits) that construct algorithms to solve the generalized conjugacy problem in hyperbolic groups.
In this talk, we give a brief introduction to relatively hyperbolic groups and present an (inefficient) algorithm that generalizes the work of Bridson and Howie to solve the generalized conjugacy problem in relatively hyperbolic groups, provided the peripheral subgroups have decidable generalized conjugacy problem. We conclude by discussing ongoing attempts to produce a more efficient (polynomial time) algorithm for relatively hyperbolic groups.