Geometric group theory Kyoto 2013,
16  19 July 2013, RIMS, Kyoto
List of talks

Daniel Allcock
Presentation of the E10 KacMoody group
KacMoody groups generalize Lie groups in the same way that Lie groups
generalize finitedimensional semisimple Lie algebras. Over the real or
complex numbers the groups are not hard to define, and with further work
Tits found a definition over any field or even ring, yielding analogues of the
Chevalley groups. His presentation is "very" infinite,
even over a finite field. For certain root systems, including the "E10" root system,
we find a simple finite presentation over any finitely generated ring, in terms of the
Dynkin diagram. Over a field a similar but lessconcrete version of this
is due to Abramenko and Muhlherr by a completely different method. Our methods
give new results even for the classical Chevalley groups. (Joint work with Lisa Carbone.)

Masayuki Asaoka
Title:
A cocycle rigidity lemma for BaumslagSolitar actions
Abstract:
We show a rigidity lemma for Diff(R^n,0)valued cocycle over
continuous actions of the BaumslagSolitar group BS(1,k).
Roughly speaking, the lemma asserts that if two cocycles
coincide up to 2jet and their linear part are good enough
then they are equivalent as cocycles.
As applications of the rigidity lemma, we also show some rigidity
results for actions of groups which contain the BS(1,k).

Mike Davis
Title: Aspherical manifolds that cannot be triangulated.
Abstract: 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 3sphere with a certain property. In 2013 Manolescu showed that such homology 3spheres do not exist; consequently, there exist ndimensional GalewskiStern manifolds that cannot be triangulated for each n > 4. By work of Freedman and Casson nontriangulable 4manifolds 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 GalewskiStern manifolds to show that there exist closed aspherical nmanifolds that cannot be triangulated for each n > 5. The question remains open in dimension 5.

Vincent Guirardel
Applications of Dehn fillings and geometric small cancellation:
an isomorphism problem, and Wise's Theorem about special quotients of
special cube complexes (MSQT)

Kate Juschenko
Title: Extensions of amenable groups by recurrent groupoids.
Abstract: I will discuss a theorem on amenability which unifies many know technical proofs of amenability to the one common proof as well as produces examples of groups for which amenability was an open problem. This is joint with V. Nekrashevych and M. de la Salle.

Nicolas Monod
Some nonamenable groups.
I will describe a very simple construction of nonamenable groups without
nonabelian free subgroups.

Shigeyuki Morita
title:
Symplectic Representations in Low Dimensional Topology
and Characteristic Classes
abstract:
We review diverse situations where representations of
the symplectic group appear in low dimensional topology.
In all of them, certain symplectic invariants give rise to
various kinds of characteristic classes. It is a very important
and difficult problem to investigate these classes.
As a possible approach to attack this problem, we analyze the structure
of the space of symplectic invariant tensors by introducing
a canonical metric on it. We then discuss certain progress
obtained in a joint project with Takuya Sakasai and Masaaki Suzuki.

Mikael Pichot
title: Triangle spaces and the Haagerup property

Kyoji Saito
Ttitle: The inversion formula for the growth of a monoid.
Abstract:
Let $M$ be a cancellative monoid and let $deg:M \to \mathbf{R}_{\ge 0}$
be a discrete degree map. Then the (left) growth function of the monoid
is defined as the generating series $P_{M,deg}(t):=\sum_{[x]\in M/\sim}
t^{deg(x)}$ of $M/\sim$: the equivalence classes of mutually left
divison relation. In the present talk, we show the inversion formula:
$P_{M,deg}(t) N_{M,deg}(t) = 1$ ,
where $N_{M,deg}(t)$ is defined as a skew generating function:
$N_{m,deg}(t):=\sum_{T} (1)^{#J(T)} \sum_{\Delta \in mcm(T)}
t^{deg(\Delta)}$ for the set $\{T\}$ of towers of (left) minimal common
multiples of irreducible elements of $M$.

Mark Sapir
Aspherical groups and manifolds with extreme properties

Zlil Sela
Title: Low dimensional topology and the elementary theory of groups
Abstract: We will survey some results on the first order theory of
free and free products of groups, and indicate concepts and
techniques from low dimensional topology that play an essential role
in proving, and at times even stating, some of these results.
These will include Tarski's problem on the elementary equivalence of
nonabelian free groups, and VaughtMalcev problem on the elementary
equivalence of free products of pairs of elementarily equivalent groups,
as well as other results on the structure of definable sets in related
theories.
We will assume no prior knowledge in model theory nor in low dimensional
topology.

Karen Vogtmann
1. An introduction to Outer spaces.
Let G be a group. The group Out(G) of outer automorphisms of G acts naturally on the set of spaces with fundamental group G, simply by changing the identification of the fundamental group with G. An outer space for G is a subset of such spaces which can be given useful topological and geometric structure. For free abelian groups, the classical symmetric space of marked flat tori is an outer space in this sense. I will discuss this as well as outer space for free groups, with some indication of what to do for groups which are partially free and partially free abelian.
2.
On the unstable homology of Out(F_n)
The natural maps Aut(F_{n1}) into Aut(F_n) and Aut(F_n) to Out(F_n) induce isomorphisms on homology in dimension i for n large with respect to i; the limiting group is the ith stable homology group. Galatius completely determined these stable homology groups, but the unstable homology remains a mystery. A breakthrough occurred when Morita found an entire series of unstable cycles, several of which have been shown to nontrivial in homology. I will discuss joint work with Jim Conant and Martin Kassabov which introduces new unstable classes and new structure in the unstable homology.

Eduardo MartinezPedroza
Title: Coherence and Negative Sectional Curvature in Complexes of Groups.
A group is coherent if finitely generated subgroups are finitely presented. We
examine a conditions on simply connected 2complexes ensuring coherence of
groups acting geometrically on them. This extends earlier work of D.Wise on
2complexes with negative sectional curvature in the case of free actions. Our
extension of this result involves a generalization of the notion of
combinatorial sectional curvature, a version of the combinatorial GaussBonnet
theorem to complexes of groups, and requires the use of L_2Betti numbers. This
is joint work with D. Wise, McGill U.