Related papers: Embeddings of braid groups into mapping class grou…
We obtain complete geometric invariants of cobordism classes of oriented simple fold maps of (n+1)-dimensional manifolds into an n-dimensional manifold N in terms of immersions with prescribed normal bundles. We compute that this cobordism…
We describe sufficient conditions which guarantee that a finite set of mapping classes generate a right-angled Artin group quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to…
In the first part of this paper we prove that the mapping class subgroups generated by the $D$-th powers of Dehn twists (with $D\geq 2$) along a sparse collection of simple closed curves on an orientable surface are right angled Artin…
We will define and study some generalisations of pure $\mathfrak{g}$-braid groups that occur in the theory of connections on curves, for any complex reductive Lie algebra $\mathfrak{g}$. They make up local pieces of the wild mapping class…
We express the rational cohomology of the unordered configuration space of a compact oriented manifold as a representation of its mapping class group in terms of a weight-decomposition of the rational cohomology of the mapping space from…
We prove that almost every interval exchange transformation, with an associated translation surface of genus $g\geq 2$, can be non-trivially and isometrically embedded in a family of piecewise isometries. In particular this proves the…
This paper is a survey of some properties of the braid groups and related groups that lead to questions on mapping class groups.
Braid groups and mapping class groups have many features in common. Similarly to the notion of inverse braid monoid inverse mapping class monoid is defined. It concerns surfaces with punctures, but among given $n$ punctures several can be…
We ask if any finite type generalized braid group is a subgroup of some classical Artin braid group. We define a natural map from a given finite type generalized braid group to a classical braid group and ask if this map is an injective…
We construct nontrivial cohomology classes of the space $Imb(S^1,\R^n)$ of imbeddings of the circle into $\R^n$, by means of Feynman diagrams. More precisely, starting from a suitable linear combination of nontrivalent diagrams, we…
We introduce a novel type of stabilization map on the configuration spaces of a graph, which increases the number of particles occupying an edge. There is an induced action on homology by the polynomial ring generated by the set of edges,…
(1) We show that if a presentation of the trivial group is "hard to trivialize", in the sense that lots of Tietze moves are necessary to transform it into the trivial presentation, then the associated presentation complex (which is a…
Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (adjacent to an angle larger than 180 degrees. In this paper we prove that the opposite statement is also true, namely that planar…
We prove that, except in certain low-complexity cases, the automorphism group of the graph of pants decompositions of a nonorientable surface is isomorphic to the mapping class group of that surface.
We describe the equivariant cohomology ring of rationally smooth projective embeddings of reductive groups. These embeddings are the projectivizations of reductive monoids. Our main result describes their equivariant cohomology in terms of…
A braided generalization of the concept of Hopf algebra (quantum group) is presented. The generalization overcomes an inherent geometrical inhomogeneity of quantum groups, in the sense of allowing completely pointless objects. All…
This paper completely determines the non-amenability of the mapping class groups of infinite-type surfaces, the mapping class groups of locally finite infinite graphs of higher ranks, gives an example of non-amenable stabiliser of a point…
We classify the orientably regular maps which are elementary abelian regular branched coverings of Platonic maps M, in the case where the covering group and the rotation group G of M have coprime orders. The method involves studying the…
The spaces of triangulations of a given manifold have been widely studied. The celebrated theorem of Pachner~\cite{Pachner} says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves, or Pachner…
Unifying various constructions of quandles including Coxeter quandles, free quandles, knot quandles of prime knots and Dehn quandles of orientable surfaces, we introduce Dehn quandles of groups with respect to their subsets. It turns out…