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Workshop 2016 "Geometry and Computation on Groups and Complexes" Abstracts
Laurent Bartholdi (ENS, Paris,
France)
Decision problems in self-similar groups
Self-similar groups are finitely generated groups "presented" by a
recursion, rather than by generators and relators. I will give
important examples, survey classical decision problems (word problem,
finiteness, conjugacy problem, order problem) and describe a partial
solution to the Engel problem, describing in particular all
ad-nilpotent elements (i.e. all y with [x,y,...,y]=1 for all x and all
long enough commutators) in the Grigorchuk group.
Nigel Boston (Wisconsin-Madison, USA)
Arboreal Galois Representations
The study of p-adic Galois representations has reaped huge rewards in
number theory and arithmetic geometry. Galois groups also naturally
act on the infinite binary rooted tree and we are developing a
parallel theory for this case. I shall describe the general context
and then go into detail regarding what is known and expected as
regards the images of these representations and of special elements,
focusing on a case involving the Basilica group.
Jacek Brodzki (Southampton,
UK)
Amenable actions of locally compact second countable
groups
The study of amenability of groups has been a very active area of
research pretty much since von Neumann introduced this notion in
1922. Among the many weaker notions introduced over the years,
exactness has gained a prominent place and has generated a lot of
interest. In the world of discrete groups, there is a deep connection
between exactness and the so called amenability at infinity, but the
transition to the locally compact case has been difficult. In this
talk I will describe how exactness in the original sense of
Kirchberg-Wassermann can be described in terms of amenable actions. In
particular, I will show that for a general locally compact second
countable group amenability at infinity is equivalent to the exactness
of the group, understood as the exactness of the reduced crossed
product functor. This is joint work with Chris Cave and Kang Li.
Inna Bumagin (Carleton, Canada)
Makanin-Razborov diagrams over relatively
hyperbolic groups
Let G be a finitely generated relatively hyperbolic group. Our goal is
to give a description of the set of homomorphisms Hom(L,G) from a
finitely generated G-limit group L to G in terms of a finite directed
rooted tree. The vertices of the tree correspond to G-limit quotients
of L and the edges correspond to epimorphisms. This is joint work
with Nicholas Touikan.
Donald Cartwright (Sydney,
Australia)
Some lattice subgroups of
PU(2,1) (slides)
A fake projective plane ("fpp") is a compact complex surface with the
same Betti numbers as the complex projective plane, but
which is not homeomorphic to it. In an important paper,
Prasad and Yeung divided the fpp's into a small number of "classes"
and found an fpp in most of these classes. Subsequently, Tim Steger
and I enumerated the fpp's, finding all 50 of them, by finding all
fpp's in each class. Part of this work involved eliminating
several``matrix algebra cases'' left open by Prasad and Yeung. This
involved a detailed study of altogether thirteen explicit lattice
subgroups of PU(2,1), and torsion-free subgroups thereof. While none
of these lattice subgroups gives rise to a fake projective plane, one
of them gives a new compact complex surface with interesting
properties. Most of the material today is a development of work with
Tim Steger, but I shall also be mentioning more recent work with
Sai-Kee Yeung and Vincent Koziarz.
Fabienne Chouraqui (Haifa,
Israel)
Knuth-Bendix algorithm and the conjugacy problems in
monoids (slides)
The use of string rewriting systems has been proved to be a very
efficient tool to solve the word problem. A question that arises
naturally is whether the use of rewriting systems may be an efficient
tool for solving other decision problems, specifically the conjugacy
problem. The complexity of this question is due to some facts, one
point is that for monoids the conjugacy problem and the word problem
are independent one of another and another point is that in semigroups
and monoids, there are several different notions of conjugacy that are
not equivalent in general. We present an algorithmic approach to the
conjugacy problems in monoids, using rewriting systems. We extend the
classical theory of rewriting systems developed by Knuth and Bendix to
a rewriting that takes into account the cyclic conjugates.
Laura Ciobanu (Neuchatel,
Switzerland)
On conjugacy growth in
groups (slides)
In this talk I will discuss various aspects of conjugacy growth in
several classes of groups, such as hyperbolic, right-angled Artin,
Baumslag-Solitar or wreath products.
Michael Farber (Queen Mary, UK)
Topology of large random spaces (slides)
I will describe several models producing large random topological
spaces, I will also present some recent results about topological
properties of such spaces (their Betti numbers, fundamental groups
etc).
Susan Hermiller (Nebraska, USA)
Trees, flow functions, and algorithms
in groups
A bounded flow function is a dynamical system on the Cayley complex of
a finitely presented group mapping the set of paths into itself, such
that path lengths increase in a bounded way and iteration eventually
maps every path into a fixed maximal tree. Although a flow function
does not imply solvability of the word problem, if the function can be
computed by a finite state automaton (FSA), the group is called
autostackable and the FSA can be used to solve the word problem for
the group. In this talk I'll discuss autostackability for closed
3-manifold groups and relatively hyperbolic groups. This includes
joint work with Mark Brittenham, Conchita Martinez-Perez, and Tim
Susse.
Derek Holt (Warwick, UK)
A new algorithmic approach to proving that
groups are hyperbolic
We describe new algorithms developed by Richard Parker, Colva
Roney-Dougal, Max Neunhoeffer, Steve Linton and others for verifying
that a group defined by a finite presentation is word-hyperbolic. They
are based on curvature arguments applied to van Kampen diagrams, and
generalise methods that have been applied to groups that satisfy small
cancellation conditions. When successful, they calculate an upper
bound for the Dehn function of the presentation, which in turn results
in an upper bound for the slim or thin triangles constant of the
Cayley graph.
In many examples it is possible to perform the calculations by
hand. This means that we can sometimes apply them to infinite families
of examples, which is an advantage over other algorithms for proving
hyperbolicity, such as the one in the speaker's KBMAG package, which
can only be applied to individual groups.
We also report on progress on an implementation of these algorithms in
GAP.
Alessandra Iozzi (ETH Zürich, Switzerland)
Irreducible lattices and bounded cohomology
We show some of the similarities and some of the differences between
irreducible lattices in product of semisimple Lie groups and their
siblings in product of locally compact groups. In the case of product
of trees, we give a concrete example with interesting properties,
among which some in terms of bounded cohomology and quasimorphisms.
Jürgen Jost (MPI MIS, Leipzig,
Germany)
On the curvature of
graphs (slides)
I shall describe some concepts for the geometric analysis of graphs
that were originally motivated by Riemannian geometry and which
provides insight into several properties of graphs that are also
useful for the analysis of empirical networks.
Riikka Kangaslampi (Aalto,
Finland)
Hyperbolic triangular buildings and periodic
apartments (slides)
This talk is motivated by Gromov's famous surface subgroup question:
Does every one-ended hyperbolic group contain a subgroup which is
isomorphic to the fundamental group of a closed surface of genus at
least 2?
We study groups acting simply transitively on the vertices of
hyperbolic triangular buildings of the smallest non-trivial
thickness. The existing examples of subgroups of groups acting on
buildings arise from periodic apartments. With the help of
computerized search we show, that most of the buildings we consider do
no have any apartments invariant under genus 2 orientable surface
group action. The existence of such an action would imply the
existence of a surface subgroup, but it is not known, whether the
existence of a surface subgroup implies the existence of a periodic
apartment. Thus, these groups are the first candidates for groups that
do not have surface subgroups arising from periodic apartments.
This is joint work with Alina Vdovina.
Ana Khukhro (Neuchatel, Switzerland)
Geometry of finite quotients of groups
The study of geometric properties of Cayley graphs of groups is known
to be extremely fruitful, often providing strong structural results
for the group. When the group is rich in finite quotients, it makes
sense to look at the geometry of the finite Cayley graphs of these
quotients, since they encode both geometric and algebraic information
about the group. Studying the connections between the geometric
properties that these quotients have uniformly, and the algebraic or
analytic properties of the parent group is not only intriguing from a
group-theoretic point of view, but can also provide us with examples
of metric spaces with interesting properties. We will focus mainly on
the relationship between the diameters and the sizes of finite
quotients, ending with some open problems.
Carsten Lange (TU Munich, Germany)
Cayley graphs of finite Coxeter groups, Colin de Verdiere matrices and convex polytopes
In 2001, Lovasz obtained an explicit description how certain weighted
adjacency matrices of a 3-connected planar graph G relate to
realizations of 3-dimensional (convex) polytopes with a vertex-edge
graph isomorphic to G. The matrices involved are known as Colin de
Verdiere matrices and they link to realizations of polytopes by
eigenvectors of the second smallest eigenvalue. In 2013, Ivrissimtzis
and Peyerimhoff studied spectral properties of transition matrices on
vertex-transitive graphs in general and, in the special case of Cayley
graphs associated to Coxeter groups in type A3,
B3 and H3, they determined the multiplicity of
the second largest eigenvalue of these transition matrices by methods
also used by Lovasz and gave a geometric interpretation in terms of
Archimedean solids.
In this talk, I review the results mentioned above and extend the
results of Lovasz and of Ivrissimtzis and Peyerimhoff to Cayley graphs
of finite Coxeter groups and associated polytopes. This presentation
is based on joint work with Ivrissimtzis, Liu and Peyerimhoff.
Shiping Liu (Durham, UK)
Higher
order Buser inequalities for the graph connection
Laplacian (slides)
In this talk, I will discuss upper bounds for eigenvalues of the
connection Laplacian on a graph with an orthogonal group or unitary
group signature. Those upper bounds are given in terms of Cheeger type
constants in the case of nonnegative Ricci curvature. This can be
considered as higher order Buser type inequalities.
This is joint work with Florentin Münch and Norbert Peyerimhoff.
Tatiana Nagnibeda (Geneva, Switzerland)
Subgroups in branch groups
The subgroup structure is one of the most basic questions in group
theory.
Branch groups can be roughly described as groups whose lattice of
subgroups contains a homogeneous rooted tree. They hence have a rich
subgroup structure.
We will discuss some of its aspects, in particular maximal and weakly
maximal subgroups and subgroup separability.
A.Yu. Olshanskii (Vanderbilt,
USA, and Moscow State University, Russia)
On the growth
of subgroups in finitely generated groups
The growth function of a subgroup H in a group G generated by a
finite set A is defined by the formula f(n) = card{ h ∈ H;
|h|A ≤ n}, where |g|A is the length of g ∈ G with respect to
A. I will discuss the behavior of growth functions in arbitrary
finitely generated groups and then focus on the growth (and co-growth)
functions of subgroups, in particular subnormal subgroups, with
respect to a free basis of a free group.
Nicolas Radu (Louvain,
Belgium)
A locally non-Desarguesian
Ã2-building admitting a uniform
lattice
An Ã2-building is a simply connected simplicial complex of
dimension 2 such that each sphere of radius 1 centered at a vertex is
isomorphic to the incidence graph of a projective plane. In 1986,
Kantor asked the problem of constructing an Ã2-building
with a cocompact lattice and whose local projective planes are finite
and non-Desarguesian. In this talk, I will explain how this problem
could be solved thanks to the work of Cartwright-Mantero-Steger-Zappa
(1993) and by making use of a computer.
Nithi Runganapirom (Frankfurt, Germany)
Quaternionic lattice of rank 2 over characteristic 2 with small quotient square complex
Although there are plenty of quaternionic arithmetic lattices of rank
2, only few of them yields a square complex as the quotient of its
action on the product of Bruhat-Tits trees with minimal number of
vertices. On the other hand, the fundamental group of a square complex
rarely has an arithmetic origin. In this talk I’m going to give a
construction of such a lattice over characteristic 2. A presentation
of such a lattice can be explicitly determined by comparing its action
on the product of Bruhat-Tits trees with the action of the
corresponding (orbital) fundamental group on the product of trees as
the universal covering of the corresponding square complex. This
lattice can be used to construct a non-classical fake quadric in
characteristic 2.
Mark Sapir (Vanderbilt, USA)
Subgroups of R. Thompson group F
Subgroups of F which have been studied for more than 25 years are
still very mysterious. For example, a seemingly random collection of
elements may generate a maximal subgroup of F or even the whole F. I
will talk about our joint work with Gili Golan and her own work which
somewhat clarifies the situation.
Vladimir Shpilrain (CUNY,
USA)
Mysteries of 2×2
matrices (slides)
We consider some special 2-generator groups and semigroups of 2×2
matrices over ℤ, ℚ, and ℤp and address various relevant algorithmic
problems and their complexity. The talk is based on joint work with
Lisa Bromberg, Anastasiia Chorna, Katherine Geller, and Alina Vdovina.
Alain Valette (Neuchatel, Switzerland)
Expanders and box spaces
Expanders, especially those coming from box spaces of residually
finite groups, have been used to test various forms of the coarse
Baum-Connes conjecture. The first construction of a pair of expanders,
one not coarsely embedding in the other, was provided by Mendel and
Naor in 2012. This was extended by Hume in 2014 who constructed a
continuum of expanders with unbounded girth, pairwise not coarsely
equivalent. In joint work with A. Khukhro, we construct a continuum of
expanders with geometric property (T) of Willett-Yu, as box spaces of
SL(3,ℤ). We will discuss the following results: if box spaces of
groups G, H are coarsely equivalent, then the groups G, H are
quasi-isometric (Khukhro and myself), and moreover G and H are
uniformly measure equivalent (K. Das).
Gerald Williams (Essex, UK)
Fibonacci-type groups and 3-manifolds
The Fibonacci groups F(2,n) are the groups defined by the
presentations with generators x0,... ,xn-1 and relations
xixi+1=xi+2 (subscripts mod n). Replacing the relations by
xixi+2=xi+1 we obtain the Sieradski groups
S(2,n). By constructing a suitable face-pairing polyhedron that
satisfies the Seifert-Threlfall condition, Sieradski proved that each
S(2,n) is a 3-manifold group. Similarly, for each even n, the
group F(2,n) is a 3-manifold group (Hilden, Lozano,
Montesinos-Amilibia; Helling, Kim, Mennicke; Cavicchioli, Spaggiari;
Howie).
The groups of Fibonacci-type G_n(m,k) are defined by presentations
with generators x0,... ,xn-1 and relations
xixi+m=xi+k, and so generalize F(2,n) and S(2,n). With
the exception of two challenging groups, we classify when the group
G_n(m,k) is a 3-manifold group. Spoiler alert: only Fibonacci groups
F(2,2m), Sieradski groups S(2,n), and cyclic groups can arise.
This is recent joint work with Jim Howie.
Dave Witte Morris (Lethbridge,
Canada)
Horospherical limit points of locally symmetric spaces
(slides)
Fix a point x in the symmetric space X associated to SL(n,ℝ). A point
z on the visual boundary of X is a "horospherical limit point" if the
SL(n,ℤ)-orbit of x intersects every horoball based at z. In the
special case of the upper half-plane model of X for n = 2, it is well
known that the horospherical limit points are precisely the irrational
numbers on the real line. For larger n, it was proved by T. Hattori
that every horospherical limit point satisfies a certain irrationality
property. We prove the converse, by applying a special case of
Ratner's Theorem on unipotent flows that was established by
S.G.Dani. Furthermore, SL(n,ℝ) can be replaced with any semisimple Lie
group and SL(n,ℤ) can be replaced with any S-arithmetic subgroup, if
we replace X with the corresponding Bruhat-Tits building. This is
joint work with G. Avramidi and K. Wortman.
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