arXiv research

Researchers compute Goeritz groups for all (1,1)-link decompositions.

Finite Goeritz groups for links with long bridge decompositions.

We combine aspects of the notions of finite decomposition complexity and asymptotic property C into a notion that we call finite APC-decomposition complexity. Any space with finite decomposition complexity has finite APC-decomposition complexity and any space with asymptotic property C has finite APC-decomposition comp…

Formula connects foliated simplicial volume with group cost.

We give an example of two JSJ decompositions of a group that are not related by conjugation, conjugation of edge-inclusions, and slide moves. This answers the question of Rips and Sela stated in "Cyclic splittings of finitely presented groups and the canonical JSJ decomposition," Ann. of Math. 146 (1997), 53-109. On th…

We find canonical decompositions for finitely presented groups which specialize to the classical JSJ-decomposition when restricted to the fundamental groups of Haken manifolds. The decompositions that we obtain are invariant under automorphisms of the group. A crucial new ingredient is the concept of a regular neighbou…

Study models of Gödel Universe using Lie groups and Iwasawa decomposition.

This is an account of the theory of JSJ decompositions of finitely generated groups, as developed in the last twenty years or so. We give a simple general definition of JSJ decompositions (or rather of their Bass-Serre trees), as maximal universally elliptic trees. In general, there is no preferred JSJ decomposition, a…

We consider the Goeritz groups of the Heegaard splittings induced from twisted book decompositions. We show that there exist Heegaard splittings of distance $2$ that have the infinite-order mapping class groups whereas that are not induced from open book decompositions. Explicit computation of those mapping class group…

Generalized Baumslag-Solitar groups are defined as fundamental groups of graphs of groups with infinite cyclic vertex and edge groups. Forester proved (in "On uniqueness of JSJ decompositions of finitely generated groups", Comment. Math. Helv. 78 (2003) pp 740-751) that in most cases the defining graphs are cyclic JSJ …

Study Goeritz groups of link decompositions, focusing on their asymptotic behavior.

Let $\mathcal{J}^1$ be the real form of a complex simple Jordan algebra such that the automorphism group is $\mathrm{F}_{4(-20)}$. By using some orbit types of $\mathrm{F}_{4(-20)}$ on $\mathcal{J}^1$, for $\mathrm{F}_{4(-20)}$, explicitly, we give the Iwasawa decomposition, the Oshima--Sekiguchi's $K_ε-$Iwasawa decomp…

We present an algorithm to construct the JSJ decomposition of one-ended hyperbolic groups which are fundamental groups of graphs of free groups with cyclic edge groups. Our algorithm runs in double exponential time, and is the first algorithm on JSJ decompositions to have an explicit time bound. Our methods are combina…

The paper finds a special pants decomposition for certain surface group representations.

Study of graphs from hexagon decompositions of surfaces.

We set up an abstract framework that allows the investigation of Iwasawa decompositions for involutive infinite-dimensional Lie groups modeled on Banach spaces. As an application, we construct Iwasawa decompositions for classical real or complex Banach-Lie groups associated with the Schatten ideals ${\mathfrak S}_p({\m…

Analogues of JSJ decompositions were developed for Poincaré duality pairs in [19]. These decompositions depend only on the group. Our focus will be on describing the edge splittings of these decompositions more precisely. We use our results to compare these decompositions with two other closely related decompositions.

The paper describes a structural decomposition of a specific type of Schottky groups.

Study homotopy groups of open books and their pages, pages, and bindings.

Classifies Morse boundaries of 3-manifold groups.

We give a description of the model theoretic relation of forking independence in terms of the notion of JSJ decompositions in non abelian free groups.

In this paper, we introduce and study various kinds of decomposition complexity. First, we give a characterization of residually finite groups having finite decomposition complexity (FDC). Secondly, we introduce equi-variant straight FDC (sFDC), and prove that a group having equi-variant sFDC if and only if its box spa…

The Hodge theorem connects cohomology groups on compact Kähler manifolds.

A function group is a finitely generated Kleinian group with an invariant connected component of its region of discontinuity. An extended function group is a finitely generated extended Kleinian group that contains orientation reversing elements and keep invariant a connected components of its region of discontinuity. …

We present an algorithm that computes Bowditch's canonical JSJ decomposition of a given one-ended hyperbolic group over its virtually cyclic subgroups. The algorithm works by identifying topological features in the boundary of the group. As a corollary we also show how to compute the JSJ decomposition of such a group o…

The study examines how automorphism growth rates of a group can be deduced from its simpler decompositions.

The pants graph of a free group is constructed and studied.

Study confined subgroups in groups with contracting elements, showing their growth rate is strictly greater than half of the ambient growth rate.

We address primary decomposition conjectures for knot concordance groups, which predict direct sum decompositions into primary parts. We show that the smooth concordance group of topologically slice knots has a large subgroup for which the conjectures are true and there are infinitely many primary parts each of which h…

This work is devoted to elaboration on the idea to use block term decomposition for group data analysis and to raise the possibility of modelling group activity with (Lr, 1) and Tucker blocks. A new generalization of block tensor decomposition was considered in application to group data analysis. Suggested approach was…

This paper and its companion arXiv:1002.4564 have been replaced by arXiv:1602.05139. We give a general simple definition of JSJ decompositions by means of a universal maximality property. The JSJ decomposition should not be viewed as a tree (which is not uniquely defined) but as a canonical deformation space of trees. …

The paper studies asymptotic dimensions of manifolds and spaces, proving key results about their geometric decompositions.

This paper studies a subgroup of the Goeritz group related to Heegaard splittings induced by openbook decompositions.

Study continuous paths in discrete subgroups of hyperbolic space, proving combination and decomposition theorems.

In this paper, we consider decompositions of basic degree 2 cohomology for a compact K-contact 5-manifold $(M,ξ,η,Φ,g)$, and conclude the pureness and fullness of $Φ$-invariant and $Φ$-anti-invariant cohomology groups. Moreover, we discuss the decomposition of the complexified basic degree 2 cohomology group. This is a…

Geometrically describes hyperbolic structures on link complements using quantum groups.

The profinite completion of the fundamental group of a closed, orientable $3$-manifold determines the Kneser--Milnor decomposition. If $M$ is irreducible, then the profinite completion determines the Jaco--Shalen--Johannson decomposition of $M$.

We study the structure of generalized Baumslag-Solitar groups from the point of view of their (usually non-unique) splittings as fundamental groups of graphs of infinite cyclic groups. We find and characterize certain decompositions of smallest complexity (`fully reduced' decompositions) and give a simplified proof of …

The orbit decomposition is given under the automorphism group on the real split Jordan algebra of all hermitian matrices of order three corresponding to any real split composition algebra, or the automorphism group on the complexification, explicitly, in terms of the cross product of H. Freudenthal and the characterist…

Abstract: Proves relative versions of group splitting results.

Decomposition complexity for metric spaces was recently introduced by Guentner, Tessera, and Yu as a natural generalization of asymptotic dimension. We prove a vanishing result for the continuously controlled algebraic K-theory of bounded geometry metric spaces with finite decomposition complexity. This leads to a proo…

Lie bialgebra structures are reviewed and investigated in terms of the double Lie algebra, of Manin- and Gauß-decompositions. The standard R-matrix in a Manin decomposition then gives rise to several Poisson structures on the correponding double group, which is investigated in great detail.

The A-B slice problem is a reformulation of the topological 4-dimensional surgery conjecture in terms of decompositions of the 4-ball and link homotopy. We show that link groups, a recently developed invariant of 4-manifolds, provide an obstruction for the class of model decompositions, introduced by M. Freedman and X.…

For all n > 0 there is a homomorphism from the smooth concordance group of knots in dimension 2n + 1 to an algebraically defined group called the rational algebraic concordance group. This algebraic concordance group splits as a direct sum of groups indexed by polynomials. For n > 1 the homomorphism is injective. This …

This paper presents a decomposition of the 3-strand Singular Braid Group.

Study simplifies homotopy groups of 6D manifolds.

We introduce the notion of a `canonical' splitting over Z or ZxZ for a finitely generated group G. We show that when G happens to be the fundamental group of an orientable Haken manifold M with incompressible boundary, then the decomposition of the group naturally obtained from canonical splittings is closely related t…

Outer automorphism group of hyperbolic groups is HHG under certain conditions.