Related papers: Intersection of subgroups in free groups and homot…

We explore the differential geometry of finite sets where the differential structure is given by a quiver rather than as more usual by a graph. In the finite group case we show that the data for such a differential calculus is described by…

The intersection growth of a group $G$ is the asymptotic behavior of the index of the intersection of all subgroups of $G$ with index at most $n$, and measures the Hausdorff dimension of $G$ in profinite metrics. We study intersection…

Let ${\cal O}$ be a quantizable coadjoint orbit of a semisimple Lie group $G$. Under certain hypotheses we prove that $#(\pi_1(\text{Ham}({\cal O})))\geq #(Z(G))$, where $\text{Ham}({\cal O})$ is the group of Hamiltonian symplectomorphisms…

We give an explicit construction of two $2$-generated subgroups $H,K\leq \SL(3,\Z)$ whose intersection is not finitely generated. The construction takes place inside the standard parabolic subgroup $\Z^2\rtimes \SL(2,\Z)\leq \SL(3,\Z)$. The…

Let G be a connected, complex reductive Lie group with maximal compact subgroup K, and let X denote the moduli space of G- or K-valued representations of a rank r free group. In this article, we develop methods for studying the…

We briefly report on our result that, if there exists a realization of a Hopf algebra $H$ in a $H$-module algebra $A$, then their cross-product is equal to the product of $A$ itself with a subalgebra isomorphic to $H$ and commuting with…

In this paper we describe the homology and cohomology of some natural bimodules over the little discs operad, whose components are configurations of non-$k$-overlapping discs. At the end we briefly explain how this algebraic structure…

We solve the subgroup intersection problem (SIP) for any RAAG G of Droms type (i.e., with defining graph not containing induced squares or paths of length 3): there is an algorithm which, given finite sets of generators for two subgroups…

In this paper, we consider the Wiener Hopf algebra, denoted $\mathcal{W}(A,P,G,\alpha)$, associated to an action of a discrete subsemigroup $P$ of a group $G$ on a $C^{*}$-algebra $A$. We show that $\mathcal{W}(A,P,G,\alpha)$ can be…

After summarising the physical approach leading to twisted homotopy and after developing the cohomological approach further with respect to our previous work we propose a third alternative approach to twisted homotopy based on group…

We compute the homotopy groups of the spaces of self maps of Lie groups of rank 2, SU(3), Sp(2), and G_2. We use the cell structures of these Lie groups and the standard methods of homotopy theory.

Given a simply connected space $X$, there are several, a priori different, algebraic groups whose groups of $\mathbb Q$-points are isomorphic to the group of homotopy classes of homotopy automorphisms of the rationalization of $X$. We will…

We study the intersection of finitely generated subgroups of free groups by utilizing the method of linear programming. We prove that if $H_1$ is a finitely generated subgroup of a free group $F$, then the WN-coefficient $\sigma(H_1)$ of…

For any algebraically closed field $K$ and any endomorphism $f$ of $\mathbb{P}^1(K)$ of degree at least 2, the automorphisms of $f$ are the M\"obius transformations that commute with $f$, and these form a finite subgroup of…

We partly generalize the estimate for the rank of intersection of subgroups in free products of groups, proved earlier by S.V.Ivanov and W.Dicks, to the case of free amalgamated products of groups with normal finite amalgamated subgroup. We…

We consider the biharmonicity condition for maps between Riemannian manifolds (see [BK]), and study the non-geodesic biharmonic curves in the Heisenberg group H_3. First we prove that all of them are helices, and then we obtain explicitly…

We show that for every subset X of a closed surface M^2 and every basepoint x_0, the natural homomorphism from the fundamental group to the first shape homotopy group, is injective. In particular, if X is a proper compact subset of M^2,…

We prove an estimate for the rank of the intersection of free subgroups in virtually free groups, which is analogous to the Hanna Neumann inequality for subgroups in a free group and to the S.V. Ivanov estimate for subgroups in free…

Let X, Y, and Z be topological modules over a topological ring $R$. In the first part of the paper, we introduce three different classes of bounded bigroup homomorphisms from $X\times Y$ into $Z$ with respect to the three different uniform…

We obtain explicit formulas for the rational homotopy groups of generalised symmetric spaces, i.e., the homogeneous spaces for which the isotropy subgroup appears as the fixed point group of some finite order automorphism of the group. In…