
Conference on
Finiteness Conditions in Topology and Algebra
Belfast, August 31September 3, 2015
Titles and abstracts of talks
Martin Bridson 
Finiteness properties, volume gradients and residually free groups
In this talk, I will describe joint work with Kochloukova in which we examine how
the number of kcells required in a minimal K(G_{n},1) grows
as one passes to subgroups of increasing finite index in a fixed group G.
These volume growth rates bound homology growth rates and are
related to L^{2} Betti numbers via Lück approximation.
We calculate the volume growth of limit groups
in all dimensions. For finitely presented residually free groups, we calculate rank gradient,
asymptotic deficiency and homology growth rates. Before outlining these calculations, I shall briefly
review the theory of limit groups and explain how the finiteness properties of residually free groups
are related to their canonical embeddings into direct products of limit groups.

Dan Burghelea 
Refinements of basic topological invariants provided by anglevalued maps
(An alternative to MorseNovikov theory)
The object of attention is the topology of a pair (compact space, degree one integral cohomology class).
One describes a class of new computable invariants associated to such pair and to an angle valued map
representing the cohomology class of the pair. One discusses their meaning, mathematical properties,
and applications/implications in mathematics and outside mathematics.

Dieter Degrijse 
Stable finiteness properties of infinite discrete groups
Let G be an infinite discrete group. A classifying space for proper actions of G is a proper
GCWcomplex X such that the fixed point sets X^{H}
are contractible for all finite subgroups H of G.
We consider the stable analogue of the classifying space for proper
actions in the category of proper Gspectra and study finiteness properties of such a
stable classifying space for proper actions. We investigate when G admits a stable
classifying space for proper actions that is finite or of finite type and relate these conditions
to classical finiteness properties of the Weyl groups of finite subgroups of G. If G is
virtually torsionfree, we show that the smallest possible dimension of a stable classifying
space for proper actions coincides with the virtual cohomological dimension of G, thus
providing a geometric interpretation of the virtual cohomological dimension of a group.

Stefan Friedl 
A marked polytope for 2generator 1relator groups
We will assign to a marked polytope to a 2generator 1relator group.
The marked points determine the BieriNeumannStrebel invariant and
the shape of the polytope itself contains information about minimal
HNNsplittings of the group. Furthermore, if the group is the
fundamental group of a 3manifold, then it determines the Thurston
norm of the 3manifold. This is joint work with Kevin Schreve and
Stephan Tillmann.

Ross Geoghegan 
The limit set of a Gmodule controlled over a GCAT(0) space
In the late 80's the BieriNeumannStrebel (BNS) invariant of a finitely generated group G appeared.
Higherdimensional analogues, as well as a version for Gmodules, were introduced a little later
by BieriRenz. These invariants have led to deep results in group theory, and they have a fundamental
relationship with tropical geometry. Technically, they are subsets of the deleted linear space V – {0},
where V:= Hom (G, R), but they are better thought of as sets of horospherical
limit points associated with the natural action of G on V. Over the last few years Robert Bieri
and I have created a substantial generalization of this theory. The linear Gspace V is
replaced by a proper CAT(0) Gspace, and our version leads to interplays with controlled topology,
arithmetic groups and their buildings, hyperbolic groups etc. A tantalizing issue is whether these ideas will
lead to a nonpositively curved version of tropical geometry. I will report and explain.

Thomas Hüttemann 
Homotopy commutative cubes, multicomplexes, and finite domination
A chain complex of Rmodules is called Rfinitely dominated if it is
homotopy equivalent to a bounded complex of finitely generated
projective Rmodules. (This notion is relevant, for example, in
topology of manifolds, group theory, and—in the guise of "perfect
complexes"—in algebraic geometry.)
An important special case can be treated explicitly: A bounded chain
complex of finitely generated free R[x,1/x]modules
is Rfinitely dominated if and only if its "Novikov homology" (homology with
coefficients in rings of formal Laurent series) is trivial (Ranicki
1995).
In the talk I will discuss a generalisation of this homological
criterion for finite domination to cover the case of Laurent
polynomial rings in many indeterminates. This is joint work with David
Quinn, and ultimately leads to a characterisation of finite domination
in terms of "projective toric varieties over noncommutative rings".
More precisely I will explain how finite domination implies triviality
of Novikov homology. The proof is inspired by a result on totalisation
of double chain complexes of Bergman. I will explain how homotopy
commutative cubical diagrams occur quite naturally in this context,
and how these are related to the formalism of multicomplexes and their
totalisations.

Claudio Llosa Isenrich
 A construction method for Kähler groups from maps onto complex tori
In this talk I will describe a method for constructing Kähler groups
F. Kähler groups are fundamental groups of compact Kähler
manifolds. The idea is to consider a fibration of a compact Kähler
manifold X over a complex torus Y. Roughly speaking we prove that
if such a fibration has isolated singularities and connected fibers
then the fundamental group F of the generic fiber (which is
Kähler ) fits into a short exact sequence 1→ F→
G → A→ 1, where G is the fundamental group of
X and A is the fundamental group of Y. This result was inspired
by work of Dimca, Papadima and Suciu: they proved it for Y a torus
of complex dimension one and used it to obtain Kähler groups with
interesting finiteness properties. It also generalises a Theorem of
Shimada.

Dessislava Kochloukova 
Weak commutativity in groups
We revise a construction of a group H = χ(G) first defined
by Sidki in 1980. The group H has a normal abelian subgroup W(G) ) with
quotient a subdirect product of three copies of G. We show some sufficient
conditions for W(G) to be finitely generated, hence of homotopical type
F_{∞}. Using results on Sigma theory or recent results on subdirect
products due to Bridson, Howie, Miller, Short this unables us to find
sufficient conditions for G to be finitely presented. We show further
that when G is a soluble group of type FP_{∞} then H is a soluble
group of type FP_{∞}. We finish with examples of soluble FP_{∞}
groups, in one example W(G) is finitely generated and in another is
infinitely generated though in all cases H is FP_{∞}. The work
presented is joint work with Saïd Sidki (University of Brasília, Brazil).

Ian Leary 
Uncountably many groups of type FP
A group G is type F if it admits a finite K(G,1). Since there
are only countably many finite group presentations, there are only countably many
groups of type F. Roughly speaking, type FP is an `algebraic shadow' of type F. In the
1990s Bestvina and Brady constructed groups that are type FP but not finitely presented.
Since BestvinaBrady groups occur as subgroups of type F groups, there are only countably
many of them. We construct uncountably many groups of type FP. As a corollary, not every
group of type FP is a subgroup of a finitely presented group.

Eduardo Martinez Pedroza 
A subgroup theorem for homological filling functions
We use algebraic techniques to study homological filling
functions of groups and their subgroups. If G is a group admitting a
finite (n+1)dimensional K(G,1) and H < G is of type
F_{n+1}, then the nth homological filling function of H is
bounded above by that of G. This contrasts with known examples where
such inequality does not hold under weaker conditions on the ambient
group G or the subgroup H. We include applications to hyperbolic
groups and homotopical filling functions. This is joint work with
Gaelan Hanlon, arXiv:1406.1046.

Alexandra Pettet 
Abstract commensurators of the Johnson filtration
The Torelli group is the subgroup of the mapping class group which acts trivially on the
homology of the surface. It is the first term of the Johnson filtration, the sequence of
subgroups which act trivially on the surface group modulo some term of its lower central
series. We prove that the abstract commensurator of each of these subgroups is the full
mapping class group. This is joint work with Martin Bridson and Juan Souto.

Tomasz Prytuła 
Classifying space for virtually cyclic stabilizers for systolic groups
Let G be a group acting properly on a systolic complex X. In this talk I will present the construction
of a finite dimensional model for the classifying space EG.
Our approach parallels the one used for CAT(0)groups by W. Lück. The key ingredient is to describe the coarse geometric
structure of a minimal displacement set of a hyperbolic isometry of X. Namely, we show that this subcomplex
of X is quasiisometric to the product of a tree and a line. This allows us to estimate the dimension of
EG from above by the dimension of X.
As a corollary we establish a conjecture of D. Wise: in a systolic group the centralizer of an element of infinite
order is commensurable with F_{n} × Z. This is joint work with D. Osajda.

Ben Quigley 
Regarding the presentation for the commutator subgroup of the Artin group of class C_{3}
We give a homological proof that this commutator subgroup is not finitely presented.
We use noncrossing partitions to construct a K(G,1) for the group. Then we use a
MayerVietoris sequence to calculate the homology of this space and in particular show that it is not
finitely generated in dimension 2. Hence via Hopf's integral homology formula the group cannot be
finitely presented.

Alexander Rahm 
Techniques for groups of finite virtual cohomological dimension
The (co)homology of the Bianchi groups has been the subject to a question by Serre, which was open
for 40 years, namely on specifying the kernel of the map induced on homology by attaching the BorelSerre
boundary to the symmetric space quotient of the Bianchi groups.
This question been given a constructive answer by the speaker. Moreover, the studies of the latter on the
(co)homology of the Bianchi groups have given rise to a new technique (called Torsion Subcomplex Reduction)
for computing the FarrellTate cohomology of discrete groups acting on suitable cell complexes.
This technique has not only already yielded general formulae for the cohomology of the tetrahedral
Coxeter groups as well as, above the virtual cohomological dimension, of the Bianchi groups
(and at odd torsion, more generally of SL_{2} groups over arbitrary number fields), it also very
recently has allowed Wendt to reach a new perspective on the Quillen conjecture; gaining structural
insights and finding a variant that can take account of all known types of counterexamples to the
Quillen conjecture. If no counterexample of completely new type surprisingly shows up, then
this refined conjecture must be valid.

Stanislav Shkarin 
Finitedimensional algebras with few relations 
Luke Steers 
Finite domination, Novikov homology and Zgraded rings
Given rings R_{0} ⊆ R and a chain complex
C of Rmodules, we say C
is Rfinitely dominated if it is a retract up to homotopy of a bounded,
finitely generated R_{0} complex.
Ranicki and later Hüttemann and Quinn proved a finite domination result for polynomial rings.
Specifically, given a Laurent polynomial ring R[x,x^{1}],
the notion of Rfinite domination of a chain
complex C of R[x,x^{1}]modules was
shown to be equivalent to C having trivial Novikov homology.
In this talk I will look at the generalisation to strongly Zgraded rings, focusing on
showing that trivial Novikov homology implies finite domination. Surprisingly, many of the ideas
that provide a proof for polynomial rings can be adapted to work for graded rings. A number of
constructions used in the polynomial case, such as quasicoherent sheaves, are redefined for
the strongly Zgraded case. In particular, this proof will also satisfy twisted polynomial
rings as a corollary.

Alex Suciu 
Sigmainvariants and tropical varieties
The BieriNeumannStrebelRenz invariants Σ^{i}(X) ⊂ H^{1}(X, R)
of a space X are the vanishing loci for the Novikov homology of X in degrees up to i.
In this talk, I will describe a connection between the Σinvariants of X and the tropicalization
of the cohomology support loci V^{i}(X) ⊂ H^{1}(X, C^{*}).

Stefan Witzel 
The Basilica Thompson group is not finitely presented
Jim Belk and Brad Forrest have constructed a group T_{B} that
acts on the Basilica Julia set much like Thompson's group T acts on
the circle. They proved that T_{B} is finitely generated and virtually
simple. I will talk about the joint result with Matt Zaremsky that T_{B}
is not finitely presented. This is an instance of the more general
problem of showing that a group G is not of type F_{n} when the (proper
geometric) dimension of G is bigger than n (infinite in this case). In
that situation local methods (like combinatorial Morse theory) seem
rarely applicable.


