arXiv research

Proves non-solvability of concordance groups using Milnor invariants.

We treat the almost differentiable left A-loops as images of global differentiable sharply transitive sections $σ:G/H \to G$ for a Lie group $G$ such that $G/H$ is a reductive homogeneous manifold. In this paper we classify all $3$-dimensional connected strongly left alternative almost differentiable left A-loops $L$, …

We investigate the existence of left-invariant closed G$_2$-structures on seven-dimensional non-solvable Lie groups, providing the first examples of this type. When the Lie algebra has trivial Levi decomposition, we show that such a structure exists only when the semisimple part is isomorphic to $\mathfrak{sl}(2,\mathb…

New findings on mapping class group actions on the circle, improving critical regularity.

Our aim in this paper is to classify the $3$-dimensional connected differentiable global Bol loops, which have a non-solvable group as the group topologically generated by their left translations and to describe their relations to metric space geometries. The classification of global differentiable Bol loops significan…

We show the existence of expanding solitons of the G$_2$-Laplacian flow on non-solvable Lie groups, and we give the first example of a steady soliton that is not an extremally Ricci pinched G$_2$-structure.

We classify all transitive actions of Lie algebras of vector fields on C^3 and R^3 up to a local equivalence and discuss why this classification can not be extended in general to the solvable case. The main technical tool is the structure of one-dimensional invariant foliations on homogeneous spaces.

New concept of metric Lie algebras helps classify Lie groups.

We construct many examples of Lie groups with compact Levi factor admitting a left-invariant metric with negative Ricci curvature. We start with a Lie algebra with Levi factor su(n) or so(n) acting on an abelian nilradical via the representation on the space of homogeneous polynomials. In the case of su(2) we obtain a …

Characterizes non-degenerate cyclic metric Lie algebras.

The paper studies acylindrical actions on trees and proves acylindrical hyperbolicity of Baumslag-Solitar groups.

According to Thurston's stability theorem, every group of C^1 diffeomorphisms of the closed interval is locally indicable (.e., every finitely generated subgroup factors through Z). We show that, even for finitely generated groups, the converse of this statement is not true. More precisely, we show that the semi-direct…

This article can be viewed as a continuation of the articles arXiv:0912.3486 and arXiv:1012.3714 where the decomposable Lie algebras admitting half-flat SU(3)-structures are classified. The new main result is the classification of the indecomposable six-dimensional Lie algebras with five-dimensional nilradical which ad…

New conditions on Riemannian manifolds' curvature tensor.

Until a couple of years ago, the only known examples of Lie groups admitting left-invariant metrics with negative Ricci curvature were either solvable or semisimple. We use a general construction from a previous article of the second named author to produce a great amount of examples with compact Levy factor. Given a c…

The paper studies knot quandles and their cohomology, proving infinite dimensionality results.

We discover a new example of a generic rank 2-distribution on a 5-manifold with a 6-dimensional transitive symmetry algebra, which is not present in Cartan's classical five variables paper. It corresponds to the Monge equation z' = y + (y'')^(1/3) with invariant quartic having root type [4], and a 6-dimensional non-sol…

An almost complex manifolds $(M^4,J)$ of real dimension 4 with non-degenerate torsion bundle admit a double absolute parallelism and it is provided the classification of homogeneous $(M^4,J)$ having an associated non-solvable Lie algebra. We extend such a classification to the analysis of the manifolds having an associ…

New link groups are derived from torus necklaces, connecting braid groups to reflection groups.

The study proves super-rigidity of Gromov's random monster group for various types of groups.

We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi--direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi--direct product of free groups. From these results we obtain a normal form of words…

Virtual twin groups map to symmetric groups, revealing automorphism structure.

Characterizes group connections on group bundles.

Study on totally symmetric sets with group applications.

Affine cactus groups are CAT(0) and hyperbolic.

The study restricts groups in graph of groups structures.

New Garside structures found for torus knot groups and related braid groups.

Logarithmic separation profile in hyperbolic groups shows hierarchical structure.

New Garside structures derived from groups, leading to new group properties.

We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups F, V, Vbr and Fbr arise from this procedure using, respectively, the systems of trivial groups, symmetric groups, braid groups and pur…

The group of 2-by-2 matrices with integer entries and determinant $\pm > 1$ can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2-dimensional torus. Thus this group is the beginning of three natural sequences of groups, name…

The study restricts normal subgroups of Kähler groups, proving specific cases and general restrictions.

New reflection groups derived from torus knots with finite meridians.

Graphically discrete groups have strong rigidity properties.

Paper proves vanishing homology groups for certain hyperbolic groups.

Study fundamental groups of geometric transformation groups using loop spaces.

Simple construction of Lie 2-groups from loop group extensions.

Study knot invariants using automorphism groups of free nilpotent groups.

We discuss dense embeddings of surface groups and fully residually free groups in topological groups. We show that a compact topological group contains a nonabelian dense free group of finite rank if and only if it contains a dense surface group. Also, we obtain a characterization of those Lie groups which admit a dens…

We exhibit a family of infinite, finitely-presented, nilpotent-by-abelian groups. Each member of this family is a solvable S-arithmetic group that is related to Baumslag-Solitar groups, and everyone of these groups has a quasi-isometry group that is virtually a product of a solvable real Lie group and a solvable p-adic…

The study shows that certain groups can be uniquely identified by their finite abelian summands.

Survey on Coxeter groups for Lie group examples.

In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various subgroups of them such as the Torelli group, the IA outer automorphism group of …

The paper studies actions on Bass-Serre trees and identifies new $C^*$-simple groups.

In this article we define the twisted product of groups as the generalization of the semidirect product of groups. We will find the necessary and sufficient condition in order that the twisted product of groups to be a group. In particular, for two copies of the same group, the twisted product of group by itself throug…

We find finite presentations for the automorphism group of the Artin pure braid group and the automorphism group of the pure braid group associated to the full monomial group.

Study the relationship between orbit braid group and equivariant mapping class group on surfaces.

The paper characterizes crystallographic groups derived from virtual braid and twin groups.