Seminaria w roku 2014/15:Wtorek, 23.06.'15, 12:15, s. 711
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.
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.
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.
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.
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.
Abstract: I will construct a map (essentially by closing braids)
F: Bn -->Conc(S3)
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
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
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)
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.
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−2 the random group Γ(n,p) is not free.
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.