Conference on Geometric Group Theory, Wrocław, June 29 — July 3, 2015


Monday Tuesday Wednesday Thursday Friday
9:00-9:50 registration
Mosher Panov Martin Garncarek
10:10-11:00 Bridson Osajda Arzhantseva Burger Okun

11:30-12:00 Nucinkis Strzałkowski Antoniuk Pawlik Scott
12:20-13:10 Caprace Farrell Leary Bożejko Przytycki
13:20 lunch
14:00-15:00 guided
city walk
15:00-15:50 Weinberger Bader Fujiwara
16:00-16:30 Riley Cordes Marcinkowski
coffee coffee
17:00-17:50 Osin Dymarz
The opening ceremony will be on Monday at 10:00.
There will be a picnic on Thursday at 18:00.

List of abstracts:

Sylwia Antoniuk

(Polish Academy of Sciences, Poland)

On the threshold for freeness of the random triangular group

We consider the model Γ(n.p) of the random triangular group, which is a group give by a random group presentation with n generators and relators of length three, where each relkator appears independently with probability p=p(n). It turns out that the behavior of the random group Γ(n.p) with respect to some interesting group properties depends strongly on the probability p and admits the so-called threshold behavior. In particular, the property of being a free group admits such a behaviour which is the main result presented in the talk. We will show that there is a constant C>0 such that for pn²<C with probability tending to 1, the random group Γ(n.p) is a free group, whereas for pn²>C with probability tending to 1 the group is not free.
This is a joint work with Tomasz Łuczak, Tomasz Prytuła, Piotr Przytycki and Bartosz Zaleski.

Goulnara Arzhantseva

(University of Vienna, Austria)

Almost implies near phenomenon in groups

We prove that the commutator is stable in permutations endowed with the Hamming distance, that is, two permutations that almost commute are near two commuting permutations. Our result extends to k-tuples of almost commuting permutations, for any given k, and allows restrictions, for instance, to even permutations.
This is a joint work with Liviu Paunescu (Bucharest). It is partially supported by my ERC grant ANALYTIC no. 259527.

Uri Bader

(Technion, Haifa, Israel)

Linearity vs. non-linearity of some automorphism groups of buildings

An "affine building" is a geometric object attached to algebraic groups over local fields by a theory developed by Bruhat and Tits in a way parallel to Cartan's more familiar theory of symmetric spaces. Nevertheless, there is a striking difference: in rank 1 and 2 one can find buildings which are not attached to any algebraic group (this never happens for symmetric spaces, nor for higher-rank affine buildings).
One then asks: for such "exotic" buildings, could the automorphism group be linear at all?
In a recent joint work with Caprace and Lecureux we give a negative answer to this question. Our method involves a combination of geometry and ergodic theory. It is built on a previous joint work with Furman regarding algebraic representation of ergodic actions.

Marek Bożejko

(University of Wroclaw, Poland)

Length functions on Coxeter groups with applications

We consider two length functions on Coxeter group (W,S) , namely |w| :=min{n : w = s1s2…sn, si∈S}, and ||w||w := min{#T : w = s1s2…sn, si∈ T⊂ S}
. We prove that they are conditionally positive definite or even we have stronger operator version. This implies that all Coxeter groups have stronger Haagerup property and infinite Coxeter groups are not Kazhdan in stronger sense so this is extension of the results of T.Januszkiewicz, R.Spatzier and myself from 1986.
We also give characterization of radial functions on (W,S) with respect the length | |, positive definite functions on the infinite permutation group and on infinite hyperoctahedral group.
Also the second length function || || is also conditionally negative definite on each Coxeter group W and this has strong consequences like:
  1. The set of generators S of the Coxeter group (W,S) is Sidon set i.e. every bounded real function on S can be extended to positive definite.
  2. The operator–Khinchine inequality holds on S.
  3. The classical normal law N(0,1) is infinitely divisible in the free probability of Voiculescu (paper in Adv.Math. by Belinschi, Bozejko, Lehner, Speicher).
The results are from the paper "Positive definite functions on coxeter groups with applications to operator spaces and noncommutative probability" by Marek Bozejko, Światosław R. Gal and Wojciech Młotkowski.

Martin Bridson

(University of Oxford, England)

Profinite rigidity for fundamental groups of low-dimensional orbifolds

In this talk I will discuss what is known about the extent to which fundamental groups of orbifolds of dimension at most 3 are determined by their finite quotients. In particular, I will explain recent work showing that the fundamental groups of punctured torus bundles can be distinguished from each other and from other 3-manifold groups by means of their profintie completions. This is joint work with Alan Reid (Texas) and Henry Wilton (Cambridge).

Marc Burger

(ETH Zurich, Switzerland)

On lattices in products of trees

Cocompact lattices in products of trees are finitely presented groups exhibiting far stranger behavior than their Lie group siblings. In this talk we report on recent work concerning their bounded cohomology; this is joint work with A.Iozzi, as well as work of A.Iozzi, A.Sisto, and C.Pagliantini.

Pierre-Emmanuel Caprace

(UC Louvain, Belgium)

From amenable CAT(0) groups to Tits buildings

The only proper CAT(0) spaces admitting a cocompact isometric action of a discrete amenable group are (quasi)flats. The goal of this talk is to describe what happens if the amenable group is allowed to be non-discrete.

Matthew Cordes

(Brandeis University, USA)

Morse boundaries of proper geodesic metic spaces

I will introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with equivalence classes of geodesic rays that identify the "hyperbolic directions" in that space. (A ray is Morse if quasi-geodesics with endpoints on the ray stay bounded distance from the ray.) This boundary is a quasi-isometry invariant and a visibility space. In the case of a proper CAT(0) space the Morse boundary generalizes the contracting boundary of Charney and Sultan and in the case of a proper Gromov hyperbolic space this boundary is the Gromov boundary. I will also discuss the Morse boundary of the mapping class group.
See: arXiv:1502.04376

Tullia Dymarz

(University of Wisconsin, Madison, USA)

Non-rectifiable Delone sets in amenable groups

In 1998 Burago-Kleiner and McMullen constructed the first examples of coarsely dense and uniformly discrete subsets of R that are not biLipschitz equivalent to the standard lattice Z. Similarly we find subsets inside the three dimensional solvable Lie group SOL that are not bilipschitz to any lattice in SOL. The techniques involve combining ideas from Burago-Kleiner with quasi-isometric rigidity results from geometric group theory.

Tom Farrell

(Tsinghua University, Beijing, China)

Bundles with extra geometric structure

This talk will report on joint work with several researchers including Pedro Ontaneda, Andrey Gogolev, Zhou Gang, Dan Knopf, Igor Belegradek and Vitali Kapovitch. We investigate smooth fiber bundles whose concrete fibers are equipped with Riemannian metrics with sectional curvatures constrained to lie in an interval I of real numbers. Prominent among these will be I=(-∞,0), (¼,1] and [0,∞).

Koji Fujiwara

(Kyoto University, Japan)

Geometry of Contracting Geodesics

One coarse feature of a hyperbolic space is that geodesics are contracting. That is if L is any geodesic then there exists B such that the nearest point projection to L of any metric ball disjoint from L has diameter at most B.
We will discuss a space with some but maybe not all geodesics are contracting, and its application to group theory. An example is a Teichmuller space. This is a joint work with Bestvina and Bromberg.

Lukasz Garncarek

(University of Wroclaw, Poland)

Boundary representations of hyperbolic groups

In my talk i will present a canonical construction of a large family of unitary representations of an arbitrary Gromov hyperbolic group. I will show that they are irreducible and inequivalent. This generalizes the results of Bader and Muchnik applying to fundamental groups of closed negatively curved manifolds.

Ian Leary

(University of Southampton, England)

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 Bestvina-Brady 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.

Michal Marcinkowski

(University of Wroclaw, Poland)

Macroscopically large rationally inessential manifolds

We construct examples of manifolds whose universal covers are large in topological sense but small in homological sense. Such manifolds provide counterexamples to a recent conjecture of A.Dranishnikov. The construction uses Coxeter groups and associated Davis complexes. A motivation to study such manifolds comes from the conjecture of Gromov. It says, that the fact that a smooth closed manifold admits a metric of positive scalar curvature is visible in the topology of its universal cover as a deficiency of the macroscopic dimension. Such a phenomenon takes place when we speak of positive sectional curvature (then the universal cover is compact). In the case of positive scalar curvature no such a result is yet known.

Alexandre Martin

(University of Vienna, Austria)

The cubical geometry of Higman's group

Higman's group was constructed as the first example of a finitely presented infinite group without non-trivial finite quotients. Despite this pathological behaviour, I will describe striking similarities with mapping class groups of hyperbolic surfaces, outer automorphisms of free groups and special linear groups over the integers. The main object of study will be the cocompact action of the group on a CAT(0) square complex naturally associated to its standard presentation. This action, which turns out to be intrinsic, can be used to explicitly compute the automorphism group and outer automorphism group of the Higman group, and to show that the group is both Hopfian and co-Hopfian. A surprisingly stronger result actually holds: Every non-trivial morphism from the Higman group to itself is an automorphism.

Lee Mosher

(Rutgers University, Newark, USA)

Hyperbolic actions and second bounded cohomology for subgroups of Out(Fn)

Motivated by a general desire to understand the large scale geometry of Out(Fn), and a specific desire to understand second bounded cohomology of subgroups of Out(Fn), we shall study various actions of subgroups of Out(Fn) on hyperbolic complexes, and the dynamics of individual elements of those actions. As an application, we shall prove that finitely generated subgroups H<Out(Fn) satisfy an alternative: either H is virtually abelian; or the second bounded cohomology of H with real coefficients is of uncountable dimension. This work is joint with Michael Handel.

Brita Nucinkis

(Royal Halloway, University of London, England)

The rational homology of generalized Richard Thompson groups

Richard Thompson's group F is defined to be the group of orientation preserving piecewise linear homeomorphisms of the unit interval with break-points dyadic integers and slopes powers of 2. Thompson also defined the groups T and V as certain groups of homeomorphisms of the circle and the Cantor-set respectively. All three groups have surprising properties. T and V are infinite, have arbitrarily high torsion, are simple and finitely presented. F is a rare example of a torsion-free group of type FP with infinite cohomological dimensions. It was shown by K. Brown that V is Q-acyclic. Here we show that for generalised Thompson groups, which turn up as automorphism groups of certain Canto-algebras, the rational homology vanishes in high dimensions. An explicit computation for the rational homology is given for Brin's group 2V. This is joint work with C. Martinez-Perez and M. Varisco.

Boris Okun

(University of Wisconsin - Milwaukee, USA)

Action dimension of right-angled Artin groups

The action dimension of a group G, actdim(G) is the least dimension of a contractible manifold which admits a proper G-action.
I will explain a partial computation of actdim for the right-angled Artin groups and why it is interesting from the point of view of -homology.
This is a joint work with Grigori Avramidi, Mike Davis, and Kevin Schreve.

Damian Osajda

(Polish Academy of Sciences and University of Wroclaw, Poland)

Groups with asymptotic asphericity properties

Free groups and small cancellation groups are well-known examples of groups that asymptotically "do not contain spheres". It is an interesting phenomenon that there are higher dimensional groups with similar properties. Among them are some weakly systolic groups, in particular, systolic groups. I will present exotic features of groups with such asphericity properties.

Denis Osin

(Vanderbilt University, USA)

Invariant random subgroups of groups acting on hyperbolic spaces.

I will survey recent results on invariant random subgroups of non-elementary isometry groups of hyperbolic spaces.

Dmitri Panov

(King's College London, England)

Ramification conjecture and complex line arrangements with aspherical complement

This talk is based on a joint work with Anton Petrunin, where we study polyhedral spaces and their ramifications. The ramification of a polyhedral space is the metric completion of the universal cover of its regular locus.
We concentrate on non-negatively curved polyhedral metric of two origins: quotients of Euclidean space by a discrete group of isometries and polyhedral metrics on CP² with singularities at a collection of complex lines. In the former case we conjecture that such quotients space always have a CAT(0) ramification and prove this in several cases. In the latter case we prove that the ramification is CAT(0) and deduce that complex line arrangements in CP² studied by Hirzebruch have aspherical complement.

Dominika Pawlik

(University of Warsaw, Poland)

Gromov boundaries of hyperbolic groups as Markov compacta

A Markov compactum (as defined by Dranishnikov) is the limit of an inverse sequence of simplicial complexes in which all the simplices can be given finite labels so that every two simplices with equal label have isomorphic pre-images (where the isomorphism should also preserve simplex labels). This property, after some subtle strengthening, enables to encode the whole inverse sequence using a finite description, where decoding resembles running a finite-state automaton. It turns out that the Gromov boundary of any finitely generated hyperbolic group admits such presentation, which gives a convenient description of the boundary and helps in designing algorithms verifying certain its topological properties. Moreover, the dimension of simplices can be bounded by the dimension of their limit. In fact, the resulting Markov compactum can be also equipped with a natural metric, quasi-conformally equivalent with the natural (i.e. Gromov visual) metric on the boundary.
It turns out that the methods used in proving these claims also allow to generalize (from the torsion-free case to all hyperbolic groups) the result of Coornaert and Papadopoulos, which provides a presentation of the boundary as a quotient of two infinite-word "regular" (in an appropriately adjusted sense) languages, which they call a semi-Markovian space.
See: D. Pawlik, Gromov boundaries of Markov compacta, preprint, 2015,

Piotr Przytycki

(McGill University, Montreal, Canada and Polish Academy of Sciences, Poland)

Arcs intersecting at most once

I will show that on a punctured oriented surface with Euler characteristic χ<0, the maximal cardinality of a set of essential simple arcs that are pairwise non-homotopic and intersecting at most once is 2|χ|(|χ|+1). This gives a cubic estimate in |χ| for a set of curves pairwise intersecting at most once, which to a great extent answers a question of Farb and Leininger.

Tim Riley

(Cornell University, USA)

Taming the hydra: the word problem and extreme integer compression

For a finitely presented group, the Word Problem asks for an algorithm which declares whether or not words on the generators represent the identity. The Dehn function is the time-complexity of a direct attack on the Word Problem by applying the defining relations.
A "hydra phenomenon" gives rise to novel groups with extremely fast growing (Ackermannian) Dehn functions. I will explain why, nevertheless, there are efficient (polynomial time) solutions to the Word Problems of these groups. The main innovation is a means of computing efficiently with compressed forms of enormous integers.
This is joint work with Will Dison and Eduard Einstein.

Rick Scott

(Santa Clara University, USA)

Growth series for Eulerian cube complexes

An compact n-dimensional nonpositively curved cube complex is Eulerian if it has the same local Euler characteristics as an n-dimensional manifold. We show that the growth series G(t) for such a cube complex satisfies the reciprocity formula G(1/t)=(-1)ⁿ G(t).

Karol Strzałkowski

(Polish Academy of Sciences, Poland)

Lipschitz simplicial volume and piecewise straightening

Lipschitz simplicial volume is a metric version of the simplicial volume, which is a homotopy invariant of manifolds with many connections with Riemannian structure. On the one hand, Lipschitz version of this invariant is more suitable for studying non-compact Riemannian manifolds of finite volume, but on the other there are fewer suitable tools to study it. One of such tools is the straightening procedure, which allows one to simplify singular chains used to compute simplicial volume, but is usually applicable only on manifolds with non-positive sectional curvature. We introduce the piecewise straightening procedure, which is a concrete generalization of the classical straightening to the case of manifolds with an upper bound on the sectional curvature. This generalization allows us to prove e.g. proportionality principle for manifolds with bounded curvature, which is a classical result for compact manifolds and states that if two Riemannian manifolds have isometric universal covers, then their Lipschitz simplicial volume is proportional to Riemannian volume.

Shmuel Weinberger

(University of Chicago, USA)

Indistinguishable manifolds

We view two manifolds as being indistinguishable, if there are uniformly contractible metrics (with a given contractibility function) on both wherein they are arbitrarily close together (in Gromov-Hausdorff space). Indistinguishable manifolds are (simple) homotopy equivalent, and have the same rational Pontrjagin classes, and cannot be distinguished by many invariants.
Many manifolds, such as spheres, lens spaces, or aspherical manifolds have no indistinguishable partners, (at least in high dimensions), as predicted by the Borel conjecture, but there are manifolds that do have these.
We shall explain this, as well as why when the fundamental group of a manifold is a lattice in a connected Real Lie group (so e.g. finite or abelian) or word hyperbolic, a manifold can have only finitely many partners. On the other hand, for a slightly modified version of a Bestvina-Brady group, there is a manifold that is indistinguishable from infinitely many others.
This is joint work with Dranishnikov and Ferry.