Geometric and Cohomological Group Theory

Special cube complexes, introduced and studied by Haglund and Wise play the key role in Wise's program that culminated in his theorem that every hyperbolic group with a quasiconvex hierarchy is residually finite. The main point is that the family of hyperbolic groups that are virtually the fundamental groups of compact special cube complexes is invariant under amalgamation and under deep quotients. We will discuss the ideas that underly that program, and describe the tools: canonical completion and cubical small cancellation.

Thompson's groups (F, T, and V) are discrete approximations to homeomorphism groups of the interval, the circle, and the Cantor set. T and V are finitely presented infinite simple groups, and F is not too far from being simple. We give a brief introduction to these groups and their (by now classical) properties. We shall discuss various ways to describe their elements diagrammatically. This is a starting point to introduce some relatives of Thompson's groups (braided versions BF and BV or analogues in higher dimensions sV) and discuss which classical properties these groups share with F, T, and V.

Classifying spaces for free actions of groups are a classical object of study. In recent years classifying spaces for actions with non-trivial stabilizers have become of increasing interest, partly because of their role played in the Baum--Connes conjecture (finite stabilizers) and the Farrell--Jones conjecture (virtually cyclic stabilizers). The finiteness properties of classifying spaces for free actions can be studied very efficiently via geometric methods using Brown's criterion. I will speak about joint work Martin Fluch providing a similar approach to classifying spaces for arbitrary families of stabilizers.

A famous theorem of Adyan--Rabin asserts that there is no algorithm to determine whether or not a finitely presented group is trivial. I'll discuss the profinite analogue of this result: there is no algorithm to determine whether or not a finitely presented group has a non-trivial subgroup of finite index. The latter property is not Markov and so, unlike many superficially similar theorems, this does not follow from the Adyan--Rabin theorem. Corollaries include a variety of new undecidability results about geometrically well behaved classes of groups, such as word-hyperbolic groups and CAT(0) groups. This is joint work with Martin Bridson.

I will give a formal sense in which Thompson's group is in fact a Ramsey-theoretic problem. The monotonicity phenomenon and growth rates discussed in the first talk fit naturally with this viewpoint and in turn suggest a natural and novel line of attack on the problem. I will discuss some partial successes along these lines and also some pitfalls.

Special cube complexes, introduced and studied by Haglund and Wise play the key role in Wise's program that culminated in his theorem that every hyperbolic group with a quasiconvex hierarchy is residually finite. The main point is that the family of hyperbolic groups that are virtually the fundamental groups of compact special cube complexes is invariant under amalgamation and under deep quotients. We will discuss the ideas that underly that program, and describe the tools: canonical completion and cubical small cancellation.

(joint work with I. Pak)

We shall present conditions under which finitely generated flat modules are projective and obtain criteria for the freeness of groups of homological dimension one.

This is a joint work with O. Kharlampovich and A. Miasnikov. We construct the first examples of finitely presented residually finite groups with arbitrarily complex (recursive) word problem, and also with easy word problem but arbitrary large (recursive) Dehn function.

Special cube complexes, introduced and studied by Haglund and Wise play the key role in Wise's program that culminated in his theorem that every hyperbolic group with a quasiconvex hierarchy is residually finite. The main point is that the family of hyperbolic groups that are virtually the fundamental groups of compact special cube complexes is invariant under amalgamation and under deep quotients. We will discuss the ideas that underly that program, and describe the tools: canonical completion and cubical small cancellation.

I shall discuss actions of arithmetic groups, Out(F) and mapping class groups on acyclic spaces, spheres and CAT(0) spaces, focussing mainly on joint work with Grunewald and Vogtmann in which we provide lower bounds on the dimension in which various arithmetic groups can act on acyclic manifolds and spheres. For example, Sp(2g,Z) has non-trivial action on an acyclic manifold of dimension less than 2g.

I will present a series of criteria (sufficient conditions) for non-amenability of groups and graphs. The applications of these criteria include establishing non-amenability of R.Thompson's group F. I will discuss major ideas of the proof.

I'll describe an ongoing project which seeks to develop new, efficient, practical means of computing with both finite and relative presentations of groups. Our methods in fact apply to a wider class of string-rewriting systems than this, and I'll start by describing our overall setup. I'll then go on to describe the way in which we have generalised small cancellation to prove concrete linear isoperimetric inequalities for a much wider class of presentations. Our approach gives rise to a wealth of new questions in pure geometric group theory, and if time permits I'll present some of these - otherwise catch me in the bar later, if you're curious!

I will give a formal sense in which Thompson's group is in fact a Ramsey-theoretic problem. The monotonicity phenomenon and growth rates discussed in the first talk fit naturally with this viewpoint and in turn suggest a natural and novel line of attack on the problem. I will discuss some partial successes along these lines and also some pitfalls.

In this talk, I will explain that the Farrell-Jones conjecture holds for all virtually solvable groups, if the semi-direct products $\Z[w,w^{-1}] \rtimes_{\cdot w} \Z$ with w a non-zero algebraic number satisfy the Farrell-Jones conjecture with finite wreath product. For the proof of the Farrell-Jones conjecture for these semi-direct products I will introduce a new method which is a combination of Farrell-Hsiang groups and transfer reducibility.

Although Kirby and Siebenmann proved that there are manifolds which do not admit PL structures, the possibility remained that all manifolds could be triangulated. In the late seventies Galewski and Stern showed that in each dimension > 4 there exist manifolds which cannot be triangulated iff there does not exist a homology 3-sphere with a certain property. In 2013 Manolescu showed that such homology 3-spheres do not exist; consequently, there exist n-dimensional Galewski-Stern manifolds that cannot be triangulated for each n > 4. By work of Freedman and Casson nontriangulable 4-manifolds exist. Are there aspherical examples of nontriangulable manifolds? In 1991 Davis and Januszkiewicz applied Gromov's hyperbolization procedure to Freedman's E_8 manifold to show that the answer is yes in dimension 4. In joint work with Jim Fowler and Jean Lafont, hyperbolization techniques are applied to the Galewski-Stern manifolds to show that there exist closed aspherical n-manifolds that cannot be triangulated for each n > 5. The question remains open in dimension 5.

Approximate schedule:

09:00 bus leaves Grey College (please collect packed lunch)

10:15 arrives at Steel Rig, walk to Housesteads (or stay on bus)

13:00 visits Housesteads and lunch

14:00 leave for Hexham

16:30 leave Hexham for Durham

17:30 arrive at Grey College

To obtain information about the cohomology of a group, one likes to have a classifying space. For F we describe such a space due to M. Stein and show how it can be used to show that F is of type $F_\infty$. We shall also indicate how the construction can be adapted to deal with the other groups in this lot (T, V, BF, BV, and sV).

Dismantlability is a property of a graph that by Polat's theorem implies existence of a clique fixed by all symmetries of the graph. Examples are diagonal graphs of CAT(0) cube complexes and 1-skeleta of systolic complexes, as well as appropriate Rips graphs for hyperbolic metric spaces. We showed with Hensel, Osajda, and Schultens that dismantlable graphs are also ubiquitous in geometric topology: the arc graph, the disc graph, the sphere graph, the Kakimizu graph. We illustrate our methods by giving a new proof of Nielsen Realisation Problem for surfaces with nonempty boundary.

We discuss the growth in homology of limit and more generaly residually free groups, as a corollary we obtain the values of analytic Betti numbers. This is a joint work with Martin Bridson.

We use algebraic methods to show that the minimal dimension of a classifying space of the mapping class group equals its virtual cohomological dimension

Dismantlability is a property of a graph that by Polat's theorem implies existence of a clique fixed by all symmetries of the graph. Examples are diagonal graphs of CAT(0) cube complexes and 1-skeleta of systolic complexes, as well as appropriate Rips graphs for hyperbolic metric spaces. We showed with Hensel, Osajda, and Schultens that dismantlable graphs are also ubiquitous in geometric topology: the arc graph, the disc graph, the sphere graph, the Kakimizu graph. We illustrate our methods by giving a new proof of Nielsen Realisation Problem for surfaces with nonempty boundary.

This continues the discussion of the 2nd lecture. M. Stein obtained a complete description of the cohomology of F by analyzing its classifying space. For the other groups, this is much more tricky since the natural geometries are not classifying spaces: the action is not free. We shall report on the state of the art and some open problems. (As far as one is only interested in finiteness properties, the action not being free is not a problem since cell stabilizers are homologically nice.)

In these talks I discuss several long-standing conjectures about Artin groups of euclidean type that have recently been resolved. My coauthors and I prove, in particular, that every irreducible euclidean Artin group is a torsion-free centerless group with a decidable word problem and a finite-dimensional classifying space. We do this by showing that it is isomorphic to a subgroup of a Garside group in the expanded sense of Digne. The Garside groups involved are introduced here for the first time. They are constructed by applying semi-standard procedures to crystallographic groups that contain euclidean Coxeter groups but which need not be generated by the reflections they contain. The first talk will discuss the relevent tools and the second will outline the proof.

Let G be a group that acts isometrically with discrete orbits on a separable CAT(0)-space X. In this talk I will discuss some Bredon finiteness properties of G for the families of finite and virtually cyclic subgroups when the space X has either finite topological dimension or the action of the group G is cocompact. As applications, I will illustrate how these results apply to the mapping class group of closed, connected, oriented surfaces and to finitely generated linear groups over fields of positive characteristic.

Let G be a finitely presented nilpotent-by-abelian-by-finite group. We show that the dimension of rational first Betti number of subgroup of finite index stabilizes. A pro-p version of the same result holds too. This is a joint work with Martin Bridson.

The groups CoCF were introduced by Holt, Rees, Röver, and Thomas, as natural generalizations of the CF groups classified circa 1985 by Muller and Schupp (with recourse to Dunwoody's accessibility theory). In 2007, Lehnert and Schwietzer showed that R. Thompson's group V is a CoCF group despite conjectures to the contrary. In Lehnert's 2008 Dissertation, he shows a group Q:= QAut(T_{2,c}) is a CoCF group and he claims an embedding of V into Q. On the strength of this, and perhaps other facts, Lehnert conjectured that Q is a universal CoCF group (any CoCF group would embed into Q, and every f.g. subgroup of Q is CoCF). In joint work with Matucci and Neunhöffer, we show that Q embeds in V (answering a question of Lehnert and Schweitzer), and also give our own proof that V embeds into Q. Thus, an upgraded version of Lehnert's conjecture is that a group is in CoCF if and only if it is fg and embeds in V. Finally, we find embeddings of other well known families of fg groups into V, showing these groups are all in CoCF as well.

This is about recent joint work with Robert Bieri. Here, G is a group, M is a proper CAT(0) space on which G acts by isometries, and A is a finitely generated G-module which is ``controlled'' over M in a sensible way that will be explained. This situation leads us to a horospherical limit set Sigma(M,A) which is a subset of the boundary-at-infinity of M. This set, and some of its cousins, are of interest in themselves, and also because they are related to diverse parts mathematics: for example: (1) buildings (when M is a suitable symmetric space), (2) tropical varieties (when G is Z^n and M is E^n), (3) geometrically finite groups of hyperbolic isometries (when M=H^n), (4) measuring the difference between the Bieri-Neumann-Strebel invariants of a group Gamma and of Gamma/A where A is an abelian normal subgroup with quotient G which is finitely presented (thus A is the right kind of G-module). (5) new properties of Thompson's Group F (still wide open). After a necessary amount of introduction I'll mainly discuss recent results of Avramedi-Morris about buildings, and of Bieri and myself about Tits metric properties.

Bruce Hughes described a class of groups that act on compact ultrametric spaces by local similarity. Thompson's group V is contained in his class, but there are many other members. All of his groups act properly on CAT(0) cubical complexes. I will describe joint work with Bruce Hughes in which we give examples of groups within his class and establish some of their properties. For instance, we describe a subclass of groups of type $F_\infty$ and some groups that are of type $F_n$ but not of type $F_{n+1}$. We also find finitely presented infinite simple groups in the class.

In the case of the group F the Sigma invariants in all dimensions are known (this is a joint work with Robert Bieri and Ross Geoghegan) and in the generalised case only in dimension 2 were recently calculated.

In these talks I discuss several long-standing conjectures about Artin groups of euclidean type that have recently been resolved. My coauthors and I prove, in particular, that every irreducible euclidean Artin group is a torsion-free centerless group with a decidable word problem and a finite-dimensional classifying space. We do this by showing that it is isomorphic to a subgroup of a Garside group in the expanded sense of Digne. The Garside groups involved are introduced here for the first time. They are constructed by applying semi-standard procedures to crystallographic groups that contain euclidean Coxeter groups but which need not be generated by the reflections they contain. The first talk will discuss the relevent tools and the second will outline the proof.

Hairy graph homology is a variant of Kontsevich's graph homology of a cyclic operad (which for appropriate operads encodes, e.g., the cohomology of mapping class groups and Out(F_n)). Hairy graph homology classes can be assembled to produce cycles for ordinary graph homology which are known to be non-trivial in the few dimensions reachable by computer. In this talk I will show how to compute hairy graph homology for graphs with cyclic fundamental group by finding a relation with Loday's dihedral homology, then using known results. This is joint work with J. Conant and M. Kassabov.