Plenary talks
Titles and abstracts
Titles and abstracts
Macarena Arenas (University of Cambridge)
Macarena Arenas (University of Cambridge)
A cubical Rips construction
A cubical Rips construction
The Rips exact sequence is a useful tool for producing examples of groups satisfying combinations of properties that are not obviously compatible. It works by taking as an input an arbitrary finitely presented group Q, and producing as an output a hyperbolic group G that maps onto Q with finitely generated kernel. The ``output group" G is crafted by adding generators and relations to a presentation of Q, in such a way that these relations create enough ``noise" in the presentation to ensure hyperbolicity. One can then lift pathological properties of Q to (some subgroup of) G. Among other things, Rips used his construction to produce the first examples of incoherent hyperbolic groups, and of hyperbolic groups with unsolvable generalised word problem.
The Rips exact sequence is a useful tool for producing examples of groups satisfying combinations of properties that are not obviously compatible. It works by taking as an input an arbitrary finitely presented group Q, and producing as an output a hyperbolic group G that maps onto Q with finitely generated kernel. The ``output group" G is crafted by adding generators and relations to a presentation of Q, in such a way that these relations create enough ``noise" in the presentation to ensure hyperbolicity. One can then lift pathological properties of Q to (some subgroup of) G. Among other things, Rips used his construction to produce the first examples of incoherent hyperbolic groups, and of hyperbolic groups with unsolvable generalised word problem.
In this talk, I will explain Rips’ result, describe a variation of it that produces cubulated hyperbolic groups of any desired cohomological dimension, and survey some tools and concepts related to these constructions, including classical and cubical small cancellation theories, cubulated groups, and asphericity.
In this talk, I will explain Rips’ result, describe a variation of it that produces cubulated hyperbolic groups of any desired cohomological dimension, and survey some tools and concepts related to these constructions, including classical and cubical small cancellation theories, cubulated groups, and asphericity.
CANCELLED Calum Ashcroft (University of Cambridge)
CANCELLED Calum Ashcroft (University of Cambridge)
Actions of random groups on cube complexes
Actions of random groups on cube complexes
In this talk I will discuss a new criterion that suffices to give an unbounded action of a group on a finite dimensional CAT(0) cube complex. In particular, this criterion provides an obstruction to Property (T). I will then discuss an application of this criterion to random groups in the Gromov density model, in particular answering a conjecture of Przytycki.
In this talk I will discuss a new criterion that suffices to give an unbounded action of a group on a finite dimensional CAT(0) cube complex. In particular, this criterion provides an obstruction to Property (T). I will then discuss an application of this criterion to random groups in the Gromov density model, in particular answering a conjecture of Przytycki.
Jim Belk (University of Glasgow)
Jim Belk (University of Glasgow)
Embeddings into Finitely Presented Simple Groups
Embeddings into Finitely Presented Simple Groups
In 1973, William Boone and Graham Higman proved that a finitely generated group G has a solvable word problem if and only if G can be embedded into a simple subgroup of a finitely presented group. They conjectured a stronger result, namely that every such group G embeds into a finitely presented simple group. This conjecture remains open after almost 50 years, but recent advances in the study of finitely presented simple groups have made it possible to verify the Boone-Higman conjecture for several large classes of groups. In this talk, I will survey results on Boone-Higman embeddings of right-angled Artin groups, countable abelian groups, contracting self-similar groups, and hyperbolic groups. This talk includes joint work with Collin Bleak, James Hyde, Francesco Matucci, and Matthew Zaremsky.
In 1973, William Boone and Graham Higman proved that a finitely generated group G has a solvable word problem if and only if G can be embedded into a simple subgroup of a finitely presented group. They conjectured a stronger result, namely that every such group G embeds into a finitely presented simple group. This conjecture remains open after almost 50 years, but recent advances in the study of finitely presented simple groups have made it possible to verify the Boone-Higman conjecture for several large classes of groups. In this talk, I will survey results on Boone-Higman embeddings of right-angled Artin groups, countable abelian groups, contracting self-similar groups, and hyperbolic groups. This talk includes joint work with Collin Bleak, James Hyde, Francesco Matucci, and Matthew Zaremsky.
Alex Evetts (University of Manchester)
Alex Evetts (University of Manchester)
Equations, formal languages, and virtually abelian groups
Equations, formal languages, and virtually abelian groups
The set of solutions to a system of equations over a group is known as an algebraic set, which is an example of a definable set (a set of tuples satisfying some sentence in the first-order language of the group). The study of algebraic sets goes back to the 1970s and 1980s and work of Makanin and Razborov on finitely generated free groups. More recently, there has been a significant amount of effort to describe algebraic sets using formal languages, and in particular the class of EDT0L languages. I will introduce this class of languages and survey some recent results, including a description of algebraic sets in virtually abelian groups (joint with A. Levine) which generalises to all definable sets (joint with L. Ciobanu).
The set of solutions to a system of equations over a group is known as an algebraic set, which is an example of a definable set (a set of tuples satisfying some sentence in the first-order language of the group). The study of algebraic sets goes back to the 1970s and 1980s and work of Makanin and Razborov on finitely generated free groups. More recently, there has been a significant amount of effort to describe algebraic sets using formal languages, and in particular the class of EDT0L languages. I will introduce this class of languages and survey some recent results, including a description of algebraic sets in virtually abelian groups (joint with A. Levine) which generalises to all definable sets (joint with L. Ciobanu).
Giles Gardam (University of Münster)
Giles Gardam (University of Münster)
The Kaplansky conjectures
The Kaplansky conjectures
There is a series of four long-standing conjectures on group rings that are attributed to Kaplansky. For example, the zero divisor conjecture states that the group ring of a torsion-free group over a field has no zero divisors. I will discuss what is known about these conjectures and my recent disproof of the unit conjecture.
There is a series of four long-standing conjectures on group rings that are attributed to Kaplansky. For example, the zero divisor conjecture states that the group ring of a torsion-free group over a field has no zero divisors. I will discuss what is known about these conjectures and my recent disproof of the unit conjecture.
Sebastian Hensel (LMU München)
Sebastian Hensel (LMU München)
Parabolics in the Fine Curve Graph
Parabolics in the Fine Curve Graph
The curve graph is a well-studied and useful tool to study 3-manifolds, and mapping class groups of surfaces. The fine curve graph is a recent variant on which the full homeomorphism group of a surface acts in an interesting way. In this talk we discuss some recent results which highlight behaviour not encountered in the "classical" curve graph. In particular, we will discuss the first entries in a dictionary between properties from surface dynamics and geometric properties of the action (and, while doing so, construct homeomorphisms acting parabolically).
The curve graph is a well-studied and useful tool to study 3-manifolds, and mapping class groups of surfaces. The fine curve graph is a recent variant on which the full homeomorphism group of a surface acts in an interesting way. In this talk we discuss some recent results which highlight behaviour not encountered in the "classical" curve graph. In particular, we will discuss the first entries in a dictionary between properties from surface dynamics and geometric properties of the action (and, while doing so, construct homeomorphisms acting parabolically).
This is joint work with Jonathan Bowden, Katie Mann, Emmanuel Militon and Richard Webb.
This is joint work with Jonathan Bowden, Katie Mann, Emmanuel Militon and Richard Webb.
Dawid Kielak (University of Oxford)
Dawid Kielak (University of Oxford)
Profinite recognition of fibring
Profinite recognition of fibring
A number of remarkable recent results in profinite rigidity use a theorem of Friedl and Vidussi that connects fibring of 3-manifolds with non-vanishing of twisted Alexander polynomials. I will discuss how a similar connection can be exhibited also in setting different from that of 3-manifolds. The talk is based on joint work with Sam Hughes.
A number of remarkable recent results in profinite rigidity use a theorem of Friedl and Vidussi that connects fibring of 3-manifolds with non-vanishing of twisted Alexander polynomials. I will discuss how a similar connection can be exhibited also in setting different from that of 3-manifolds. The talk is based on joint work with Sam Hughes.
Marco Linton (University of Warwick)
Marco Linton (University of Warwick)
Simplifying Gersten's conjecture
Simplifying Gersten's conjecture
One-relator groups with exceptional intersection were first studied by Collins and Howie twenty years ago. In this talk, I will show that such groups have negative immersions if and only if they contain no Baumslag--Solitar subgroups. Hence, by a recent result of mine, such groups satisfy Gersten's conjecture; that is, they are hyperbolic if and only if they contain no Baumslag--Solitar subgroups. I will then introduce two new families of one-relator groups and show how we can use this to reduce Gersten's conjecture to these two families.
One-relator groups with exceptional intersection were first studied by Collins and Howie twenty years ago. In this talk, I will show that such groups have negative immersions if and only if they contain no Baumslag--Solitar subgroups. Hence, by a recent result of mine, such groups satisfy Gersten's conjecture; that is, they are hyperbolic if and only if they contain no Baumslag--Solitar subgroups. I will then introduce two new families of one-relator groups and show how we can use this to reduce Gersten's conjecture to these two families.
Dan Margalit (Georgia Tech)
Dan Margalit (Georgia Tech)
Fast Nielsen--Thurston Classification
Fast Nielsen--Thurston Classification
Each element of the mapping class group has one of three types: periodic, pseudo-Anosov, or reducible. In joint work with Strenner, Taylor, and Yurttas, we give an algorithm to determine which. Our algorithm is quadratic with respect to word length in the mapping class group. A polynomial time algorithm was previously given by Bell and Webb. In this talk we will explain the piecewise linear action of the mapping class group on the space of measured foliations, and how we use piecewise-linear algebra to determine the type of a mapping class.
Each element of the mapping class group has one of three types: periodic, pseudo-Anosov, or reducible. In joint work with Strenner, Taylor, and Yurttas, we give an algorithm to determine which. Our algorithm is quadratic with respect to word length in the mapping class group. A polynomial time algorithm was previously given by Bell and Webb. In this talk we will explain the piecewise linear action of the mapping class group on the space of measured foliations, and how we use piecewise-linear algebra to determine the type of a mapping class.
Jean Pierre Mutanguha (Princeton)
Jean Pierre Mutanguha (Princeton)
Canonical forms for free group automorphisms
Canonical forms for free group automorphisms
The Nielsen–Thurston theory of surface homeomorphisms can be thought of as a surface analogue to the Jordan Canonical Form. I will discuss my progress in developing a similar canonical form for free group automorphisms. (Un)Fortunately, free group automorphisms can have arbitrarily complicated behaviour. This is a significant barrier to translating arguments that worked for surfaces into the free group setting; nevertheless, the overall ideas/strategies do translate!
The Nielsen–Thurston theory of surface homeomorphisms can be thought of as a surface analogue to the Jordan Canonical Form. I will discuss my progress in developing a similar canonical form for free group automorphisms. (Un)Fortunately, free group automorphisms can have arbitrarily complicated behaviour. This is a significant barrier to translating arguments that worked for surfaces into the free group setting; nevertheless, the overall ideas/strategies do translate!
Alina Vdovina (University of Newcastle)
Alina Vdovina (University of Newcastle)
Classifying new higher-dimensional analogues of the Thompson groups by K-theory of C*-algebras.
Classifying new higher-dimensional analogues of the Thompson groups by K-theory of C*-algebras.
We present explicit constructions of infinite families of CW-complexes of arbitrary dimension with buildings as the universal covers. These complexes give rise to new families of C*-algebras, classifiable by their K-theory. The underlying building structure allows explicit computation of the K-theory. We will present new higher-dimensional generalizations of the Thompson groups, which are usually difficult to distinguish, but the K-theory of C*-algebras gives new invariants to recognize non-isomorphic groups.
We present explicit constructions of infinite families of CW-complexes of arbitrary dimension with buildings as the universal covers. These complexes give rise to new families of C*-algebras, classifiable by their K-theory. The underlying building structure allows explicit computation of the K-theory. We will present new higher-dimensional generalizations of the Thompson groups, which are usually difficult to distinguish, but the K-theory of C*-algebras gives new invariants to recognize non-isomorphic groups.
Yvon Verberne (University of Toronto)
Yvon Verberne (University of Toronto)
Automorphisms of the fine curve graph
Automorphisms of the fine curve graph
The fine curve graph of a surface was introduced by Bowden, Hensel and Webb. It is defined as the simplicial complex where vertices are essential simple closed curves in the surface and the edges are pairs of disjoint curves. We show that the group of automorphisms of the fine curve graph is isomorphic to the group of homeomorphisms of the surface, which shows that the fine curve graph is a combinatorial tool for studying the group of homeomorphisms of a surface. This work is joint with Adele Long, Dan Margalit, Anna Pham, and Claudia Yao.
The fine curve graph of a surface was introduced by Bowden, Hensel and Webb. It is defined as the simplicial complex where vertices are essential simple closed curves in the surface and the edges are pairs of disjoint curves. We show that the group of automorphisms of the fine curve graph is isomorphic to the group of homeomorphisms of the surface, which shows that the fine curve graph is a combinatorial tool for studying the group of homeomorphisms of a surface. This work is joint with Adele Long, Dan Margalit, Anna Pham, and Claudia Yao.
Karen Vogtmann (University of Warwick)
Karen Vogtmann (University of Warwick)
The integral Euler characteristic of Out(F_n)
The integral Euler characteristic of Out(F_n)
This is joint work with Michi Borinsky. The alternating sum of the Betti numbers of a group G is sometimes called the integral Euler characteristic e(G), to distinguish it from the rational Euler characteristic \chi(G) introduced by C.T.C. Wall. The rational Euler characteristic is easier to compute but does not necessarily say anything about the Betti numbers…for instance the rational Euler characteristic of GL(n,Z) is zero in spite of the fact that GL(n,Z) has a lot of non-trivial homology. I will explain recent results on the asymptotic behavior of the integral Euler characteristic of the group Out(F_n) of outer automorphisms of a free group. This extends our previous work on the rational Euler characteristic of Out(F_n), and uses results from the 1990’s about fixed point sets of finite-order elements of Out(F_n) acting on Outer space.
This is joint work with Michi Borinsky. The alternating sum of the Betti numbers of a group G is sometimes called the integral Euler characteristic e(G), to distinguish it from the rational Euler characteristic \chi(G) introduced by C.T.C. Wall. The rational Euler characteristic is easier to compute but does not necessarily say anything about the Betti numbers…for instance the rational Euler characteristic of GL(n,Z) is zero in spite of the fact that GL(n,Z) has a lot of non-trivial homology. I will explain recent results on the asymptotic behavior of the integral Euler characteristic of the group Out(F_n) of outer automorphisms of a free group. This extends our previous work on the rational Euler characteristic of Out(F_n), and uses results from the 1990’s about fixed point sets of finite-order elements of Out(F_n) acting on Outer space.
Ric Wade (University of Oxford)
Ric Wade (University of Oxford)
Direct products of free groups in ${{\rm{Aut}}}(F_N)$
Direct products of free groups in ${{\rm{Aut}}}(F_N)$
We give a complete description of the embeddings of direct products of nonabelian free groups into ${{\rm{Aut}}}(F_N)$ and ${{\rm{Out}}}(F_N)$ when the number of direct factors is maximal. To achieve this, we prove that the image of each such embedding has a canonical fixed point of a particular type in the boundary of Outer space. Joint work with Martin Bridson.
We give a complete description of the embeddings of direct products of nonabelian free groups into ${{\rm{Aut}}}(F_N)$ and ${{\rm{Out}}}(F_N)$ when the number of direct factors is maximal. To achieve this, we prove that the image of each such embedding has a canonical fixed point of a particular type in the boundary of Outer space. Joint work with Martin Bridson.