## Seminaria w roku 2014/15:

Wtorek, 23.06.'15, 12:15, s. 711Poniedziałek, 11.05.'15, 14:15, s. 711 Wtorek, 12.05.'15, 8:15, s. 711 Poniedziałek, 13.04.'15, 14:15, s. 711

**Antoine Clais**(U. Lille 1):

*Combinatorial modulus on boundaries of some right-angled hyperbolic buildings*Abstract

**Jacek Świątkowski**(UWr.):

*n-wymiarowe kompakty Sierpinskiego jako brzegi prostokatnych grup Coxetera*Abstract

**Jérémie Chalopin**(LIF Marseille):

*Cop and robber game and hyperbolicity*Abstract: In this talk, we consider a variant of the cop and rober game where the cop and the robber move at different speed. The difference with the classical cop and robber game is that at each step, the cop can move along a path of length at most s', and the robber can move along a path of length at most s without going through the position of the cop. A graph is (s,s')-copwin if the cop with speed s' has a strategy to capture any robber moving at speed s.

We will present some results relating the cop and robber game and Gromov hyperbolicity of a graph. We show that if a graph is δ-hyperbolic, then it is (2r,r+2δ)-copwin for any r. Conversely, we show that a (s,s')-copwin graph is δ-hyperbolic with δ = O(s

^{2}). From our approach, we deduce an O(n

^{2}) algorithm to approximate the hyperbolicity of a graph when we are given its distance matrix.

This talk is based on joint works with V. Chepoi, N. Nisse and Y. Vaxes, and with V. Chepoi, P. Papasoglu and T. Pecatte

**Jan Czajkowski**(IMPAN):

*Własnosci włoknistych klastrow w perkolacji Bernoulliego*

Streszczenie: Chciałbym przedstawic pewne moje pomysły na badanie
własnosci perkolacji Bernoulliego w fazie niejednoznacznosci z
parametrem blisko prawdopodobienstwa krytycznego. Korzystam z faktu, ze
wowas niesko.cnone klastry maja strukture ,drzewa skonczonych
grafow. Dwie własnosci, kto zamierzam rozwazac, to:

- Dla dowolnego wierzchołka x grafu prawdopodobienstwo, ze dany
wierzchołek y znajdzie sie w jednym klastrze wraz z x, maleje do 0 wraz
z odległoscia y od x.

- Dla przypadku grafohiperbolicznych: p.n. dowolna nieskonczona
sciezka w klastrze zbiega do pewnego punktu w brzegu Gromowa grafu.

**Ioana-Claudia Lazar**(U. Timisoara):

*A combinatorial negative curvature condition*

Abstract: Recently, Damian Osajda introduced a combinatorial curvature condition called m-location (m > 6) for flag simplicial complexes. According to Osajda's definition, a flag simplicial complex is m-located if every so called dwheel with the boundary length at most m is contained in a 1-ball. We shall enlarge this definition of m-location. Namely, we require that any homotopically trivial loop of length at most m should be contained in the link of a vertex. Using this more general definition of m-location, we shall extend some of Osajda's results. We prove that the universal cover of an 8-located simplicial complex is an 8-located simplicial complex satisfying the SD' property. The SD' property is a global combinatorial condition on flag simplicial complexes. Eventually, we show that a simply connected 8-located complex is hyperbolic.

**Michał Kukieła**(UMK):

*Fixed points in dismantlable, rayless posets*Abstract: We will recall the notion of dismantability of a poset (with respect to a class of retractions) and some of its applications in fixed point theory of order-preserving maps. Connections to homotopy theory will also be mentioned. Next we will generalize some of those results from finite to rayless posets, proving in particular that if

*P*is a rayless poset dismantlable to a point, then for any group

*G*acting on

*P*and any order-preserving endomorphism

*f: P-->P*the sets of fixed points

*P*and

^{G}*Fix(f)*are both dismantlable. Similar results will be given for simplicial complexes. On the way we will discuss codismantlability, local dismantlability and related notions, and state some open problems.

**Jarosław Grytczuk**(UJ):

*Nonrepetitive coloring of graphs*

Abstract: A coloring of a graph *G* is nonrepetitive if no simple path of
*G *contains two identical blocks of colors in a row. In 1906 Thue proved
that three colors are sufficient for such coloring of an infinite path.
This result is the starting point of Combinatorics on Words - a wide
discipline with lots of exciting problems, results, and applications
(for instance in a famous Burnside problem for finitely generated groups
of bounded exponent). In the talk I will present some recent
developments in graph theoretic part of this area. Most of them
concentrate around the main conjecture stating that planar graphs are
nonrepetitively colorable with some constant number of colors.

**Matthias Blank**(U. Regensburg):

*Relative bounded cohomology via groupoids*

Abstract: Bounded cohomology is a functional analytic modification of regular cohomology, with applications to geometry, topology and (geometric) group theory. After giving a very short introduction to groupoids, I will present our construction of (relative) bounded cohomology for (pairs) of groupoids. This includes a natural setting for bounded cohomology relative to a family of subgroups. Finally, I will discuss a relative version of Gromov's mapping Theorem in this context.

**Dominika Pawlik**(UW):

*Gromov boundaries of hyperbolic groups as inverse limits of polyhedra*

Abstract: Hyperbolic groups are known for their automaticity, proved by Cannon, which also leads to certain regularity properties of their Gromov boundaries. It turns out that this regularity can be made 'simplicial' by presenting the boundary of any such group (up to a homeomorphism) as the inverse limit of the system of nerves of its certain covers, constructed so that the nerves satisfy Markov property (as defined by Dranishnikov); in addition, the dimension of these nerves can be bounded by the dimension of their limit. In fact, the inverse limit of such system 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 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.

**Jarek Kedra**(U. Aberdeen):

*A quasihomomorphism from braids to concordance classes of knots*

Abstract: I will construct a map (essentially by closing braids)
F: B_{n} -->Conc(S^{3})
from the braid group to the concordance group of knots in the three
dimensional sphere and prove that this map has good algebraic and
geometric properties. Namely, it is a quasihomomorphism with respect to
the slice genus and it is Lipschitz with respect to the biinvariant word
metric on the braid group and the slice genus on the concordance
group.

As an application I will construct infinite families of knots (and
concodrance classes) with uniformly bounded slice genus and infinite
sequences of concordance classes with growing four ball genus.

Joint work with Michael Brandenbursky

**Alexandre Martin**(U. Wien):

*Cubulation of small cancellation groups over free products*

Abstract: We know since work of Wise that *C’(1/6)* small cancellation groups are
cubulable, i.e. they act properly and cocompactly on a CAT(0) cube complex. Certain hyperbolic groups, although not small cancellation groups in the classical sense,
can be seen as *C’(1/6)* small cancellation groups over a free product of cubulable
groups. In this talk, I will present a cubulation theorem for *C’(1/6)* small cancellation groups over a free product of ﬁnitely many cubulable groups. Such groups
act very nicely on a small cancellation polygonal complex with cubulable vertex
stabilisers. I will explain how one can ”combine” the various wallspaces structures
(on vertex stabilisers, on the polygonal complex) into a wallspace structure for the
whole group. This is joint work with M. Steenbock (University of Vienna)

**Michał Marcinkowski:**

*Macroscopically large homologically small manifolds*

Abstract: I will give the first examples of rationally inessential but macroscopically large manifolds. Such manifolds are counterexamples to the Dranishnikov rationality conjecture. In some cases we are able to prove that they do not admit a metric of positive scalar curvature, thus support the Gromov positive scalar curvature conjecture. Fundamental groups of these manifolds are right angled Coxeter groups. The construction uses small covers of convex polyhedrons (or alternatively Davis complexes) and surgery.

**Sylwia Antoniuk**(IMPAN):

*On the threshold when the random triangular group is no longer free*

Abstract: We consider the binomial model *Γ(n,p)* of a random triangular group,
in which the group is given by a random presentation *⟨S|R⟩* with *n* generators and
relators of length three, such that each relator is present in *R* independently with
probability *p*. We are interested in the asymptotic behavior of the random group
*Γ(n,p)* when *n* goes to infinity and *p=p(n)*. In particular, we show that there
exists a constant *c>0* such that for any *ε>0*, with probability tending to *1*, if
*p<(c−ε)n ^{−2}* then

*Γ(n,p)*is a free group, whereas for

*p≥(c+ε)n*the random group

^{−2}*Γ(n,p)*is not free.

**Damian Osajda**(IMPAN):

*Orthoscheme complexes of modular lattices are CAT(0)*

Abstract: Tom Brady and Jon McCammond associated a metric simplicial complex—the orthoscheme complex—with every graded poset. This is related to their work on the CAT(0) property for braid groups. They conjectured that the orthoscheme complex of a modular lattice is CAT(0). I will present the proof of this statement from a recent joint paper with Jérémie Chalopin, Victor Chepoi and Hiroshi Hirai.