arXiv research

Study explores associated groups of symmetric quandles and their properties.

Computes quandle associated groups using group homology.

The study shows acylindrical hyperbolicity for Artin groups not associated with joins or cones.

The paper classifies Lorentzian Lie groups based on Codazzi tensors and quasi-statistical structures.

Drinfel'd used associators to construct families of universal representations of braid groups. We consider semi-associators (i.e., we drop the pentagonal axiom and impose a normalization in degree one). We show that the process may be reversed, to obtain semi-associators from universal representations of 3-braids. We v…

New mechanics on non-associative octonions discovered.

The study proves conjecture for specific Artin groups.

In this paper, we compute the Wanas tensor associated to canonical connections on three-dimensional Lorentzian Lie groups with some product structure. We define algebraic Wanas solitons associated to canonical connections. We classify algebraic Wanas solitons associated to canonical connections on three-dimensional Lor…

The paper calculates Alexander polynomials for knots using finite group representations.

We prove Patterson's conjecture about the singularities of the Selberg zeta function associated to a convex-cocompact, torsion free group acting on a hyperbolic space.

The study classifies special flows on specific geometric groups.

Investigate local Lie group structure of bisections over compact manifolds

Classifies Ricci collineations on specific 3D Lorentzian Lie groups.

Study finds all conformal Ricci collineations on specific 3D Lorentzian groups.

We prove that the Waldhausen Nil-group associated to a virtually cyclic groups that surjects onto the infinite dihedral group vanishes if and only if the corresponding Farrell Nil-group associated to the canonical index two subgroup is trivial. The proof uses the transfer map to establish one direction, and uses contro…

Study equidistribution for flows on geometrically finite convergence group actions.

In this paper we study unitary braid group representations associated with Majorana Fermions. Majorana Fermions are represented by Majorana operators, elements of a Clifford algebra. The paper recalls and proves a general result about braid group representations associated with Clifford algebras, and compares this resu…

Study on string links invariant under associator choice and Grothendieck--Teichmüller group action.

Study Veech groups in fibered 3-manifolds, proving no parabolics for fibers.

We consider a family of 2-step nilpotent Lie algebras associated to uniform complete graphs on odd number of vertices. We prove that the symmetry group of such a graph is the holomorph of the additive cyclic group $\Z_n$. Moreover, we prove that the (Lie) automorphism group of the corresponding nilpotent Lie algebra co…

We compute the automorphism groups of the Dolbeault, de Rham and Betti moduli spaces for the multiplicative group ${\mathbb C}^*$ associated to a compact connected Riemann surface.

The paper classifies Ricci solitons on specific Lorentzian Lie groups.

The paper calculates curvature limits and Gauss-Bonnet theorems in the Heisenberg group.

Artin groups not free of infinity are shown to have finite centers.

Study finds all Ricci collineations for specific connections on 3D Lorentzian groups.

Building on ideas from [DT98; DS11; Wal17; Hay17], we outline a proposal for constructing Floer homology groups associated with a G2-manifold. These groups are generated by associative submanifolds and solutions of the ADHM Seiberg-Witten equations. The construction is motivated by the analysis of various transitions w…

To any semigroup presentation $\mathcal{P}= \langle Σ\mid \mathcal{R} \rangle$ and base word $w \in Σ^+$ may be associated a nonpositively curved cube complex $S(\mathcal{P},w)$, called a Squier complex, whose underlying graph consists of the words of $Σ^+$ equal to $w$ modulo $\mathcal{P}$ where two such words are lin…

The paper derives Gauss-Bonnet theorems for deformed connections in affine and rigid motions groups.

Positive braids linked to knot invariants and geometric monodromy groups.

We investigate the planarity of the boundaries of right-angled Coxeter groups. We show that non-planarity of the defining graph does not necessarily imply non-planarity of every boundary of the associated right-angled Coxeter group, although it does in many cases. Our techniques yield a characterization of the triangle…

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.

Our goal is to better understand the relationship between the polyhedron and the group associated with a fundamental domain in H^3. In this paper, we will study torsion-free groups and determine a formula for how many edge classes a given abstract polyhedron must have. We will use that result to classify all fundamenta…

Characterizes group connections on group bundles.

We compute the log canonical thresholds of non-negatively curved singular hermitian metrics on ample linearized line bundles on bi-equivariant group compactifications of complex reductive groups. To this end, we associate to any such metric a convex function whose asymptotic behavior determines the log canonical thresh…

We compute the equivariant $K$-homology of the classifying space for proper actions, for compact 3-dimensional hyperbolic reflection groups. This coincides with the topological $K$-theory of the reduced $C^\ast$-algebra associated to the group, via the Baum-Connes conjecture. We show that, for any such reflection group…

In this thesis, we introduce a new cohomology theory associated to a Lie 2-algebras and a new cohomology theory associated to a Lie 2-group. These cohomology theories are shown to extend the classical cohomology theories of Lie algebras and Lie groups in that their second groups classify extensions. We use this fact to…

Develops Patterson-Sullivan theory for coarse cocycles.

Three new types of graded Lie groups are constructed and analyzed.

Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.

We analyze the degree-two part of the Torelli group's associated graded.

Semisimplicity proven for conformal blocks representations.

Defines and computes geometric pairings for discrete groups using Baum-Connes assembly map.

We show that if K is a non-trivial knot inside a homology sphere X, the rank of the knot Floer homology group associated with K is strictly bigger than the rank of the Heegaard Floer homology group associated with X.

In this paper, we compute canonical connections and Kobayashi-Nomizu connections and their curvature on three-dimensional Lorentzian Lie groups with some product structure. We define algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections. We classify algebraic Ricci solitons assoc…

A generalized Stiefel manifold is the manifold of orthonormal frames in a vector space with a non-degenerated bilinear or hermitian form. In this article, the Isometry group of the generalized Stiefel manifolds are computed at least up to connected components in an explicit form. This is done by considering a natural n…

The affine diffeomorphism group $\mathrm{Aff}(S,q)$ of a half-translation surface $(S,q)$ comprise the self-diffeomorphisms with constant differential away from the singularities. This group coincides with the stabiliser of the associated Teichmüller disc under the action of the mapping class group on Teichmüller space…

Residual finiteness is known to be an important property of groups appearing in combinatorial group theory and low dimensional topology. In a recent work [2] residual finiteness of quandles was introduced, and it was proved that free quandles and knot quandles are residually finite. In this paper, we extend these resul…

We give explicit descriptions of the adjoint group of the Coxeter quandle $Q_W$ associated with an arbitrary Coxeter group $W$. The adjoint group of $Q_W$ turns out to be an intermediate group between $W$ and the corresponding Artin group $A_W$, and fits into a central extension of $W$ by a finitely generated free abel…