arXiv research

Introduces $q$-transpose for $q$-deformed modular group matrices.

Finite specializations of a q-deformed modular group at roots of unity.

Study of $q$-rationals and their geometric properties, including deformed Farey triangulation and Springborn operations.

The paper discusses $q$-deformations of the Aomoto complex.

Compactifies stability space for $A_2$ category, introducing $q$-deformed rational numbers.

Link between braid groups and q-deformed rationals solves a classification problem.

We discuss an approach to quantum gerbes over quantum groups in terms of q-deformation of transition functions for a loop group bundle. The case of the quantum group SUq(2) is treated in some detail.

New q-deformed integers help compute Jones polynomials efficiently.

The paper constructs TQFTs and Schrödinger representations for Heisenberg group.

Proves existence and uniqueness of loop equation solutions for semisimple Frobenius manifolds.

Countable modular groups found on surfaces with infinite type.

We show that the modular group has an infinite family of finite index subgroups, each of which has the same trace set as the modular group itself. Various congruence subgroups of the modular group, and the Bianchi groups, are also shown to have this property. In the case of the modular group, we construct examples of s…

Picard modular groups are shown to be generated by complex reflections.

Researchers found the global topology of the Eisenstein-Picard modular surface.

We give a method to compute presentations of saturated cluster modular groups. Using this, we obtain finite presentations of the saturated cluster modular groups of finite mutation type $X_6$ and $X_7$. We verify that the cluster modular groups of finite mutation type $\widetilde{E}_6$, $\widetilde{E}_7$, $\widetilde{E…

Study inert and ambiguous classes in modular group using combinatorial methods.

Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle groups or to Teichmueller curves. In Part I of this paper we study the arithmetic properties of the modular embedding and deve…

Unified treatment of gauge theories and Yang-Mills theory duality.

Method constructs fundamental domains for Picard modular groups.

We give formulas for the Whitehead groups and the rational $K$-theory groups of the (integer group ring of the) Hilbert modular group in terms of its maximal finite subgroups.

Quantum theory uses modular group representations to assign invariants to 3-manifolds.

New anomaly cancellation formulas for E8*E8*E8 gauge group.

We investigate modular embeddings for semi-arithmetic Fuchsian groups. First we prove some purely algebro-geometric or even topological criteria for a regular map from a smooth complex curve to a quaternionic Shimura variety to be covered by a modular embedding. Then we set up an adelic formalism for modular embeddings…

Abstract: Mapping class groups act on cohomology of surfaces via Hochschild cohomology.

New Fuchsian groups found with special embedding properties.

Growth rates of geodesics on modular orbifolds are studied.

We construct families of TQFT's over the finite field Z/pZ starting from an integral TQFT obtained by Frohman and Nicas. These TQFT's are likely to describe the constant order contributions of the cyclotomic integer expansions of the Reshetikhin Turaev Ohtsuki theories. Their modular structure is intimately related to …

The paper identifies a component of representations mapping modular group elements to isometries with unique fixed points.

Fixed points found in cluster modular groups under specific conditions.

Study Eisenstein metrics on modular group representations.

In this paper we provide descriptions of the Whitehead groups with coefficients in a ring of the Hilbert modular group and its reduced version, as well as for the topological K-theory of $C^*$-algebras, after tensoring with $\mathbb{Q}$, by computing the source of the assembly maps in the Farrell-Jones and the Baum-Con…

We characterize the semi-conjugacy class of a Fuchsian action of the modular group on the circle in terms of rotation numbers of two standard generators and that of their product. We also show that among lifts of a Fuchsian action of the modular group, only 5-fold lift admits a similar characterization. These results i…

We study the asymptotic behaviour of the quantum representations of the modular group in the large level limit. We prove that each element of the modular group acts as a Fourier integral operator. This provides a link between the classical and quantum Chern-Simons theories for the torus. From this result we deduce the …

Linking numbers of modular knots derived from geometric and algebraic properties.

The article is devoted to the $q_R$-conformal modular functors, which being ``deformations'' of the conformal modular functor (the projective representation of the category $Train(Diff_+(S^1))$, the train of the group $Diff_+(S^1)$ of all orientation preserving diffeomorphisms of a circle) in the class of all projectiv…

Geometrically, Legendrian surfaces related by surgery have related skein-valued cluster spaces.

A hyperbolic conjugacy class in the modular group PSL(2,Z) corresponds to a closed geodesic in the modular orbifold. Some of these geodesics virtually bound immersed surfaces, and some do not; the distinction is related to the polyhedral structure in the unit ball of the stable commutator length norm. We prove the foll…

Classical Dedekind sums are connected to the modular group through the construction of a (Dedekind) symbol on the cusp set of the modular group. In this paper we study generalizations of Dedekind symbols and sums that can be associated to certain Fuchsian groups uniformizing 1-punctured tori.

In this paper we use fractal geometry to investigate boundary aspects of the first homology group for finite coverings of the modular surface. We obtain a complete description of algebraically invisible parts of this homology group. More precisely, we first show that for any modular subgroup the geodesic forward dynami…

In this paper, we extend the notion of modular functor and fusion category to what we called $G$ equivariant modular functor and $G$ equivariant fusion category, where $G$ is a finite group, and establish a correspondence between between these notions.

Describes representations of modular group into SL(3,R)/SO(3).

The paper determines modular cohomotopy groups up to extensions using classical and unstable homotopy methods.

In this note, we fill in a gap in the literature by proving that the Teichmueller modular groups (mapping class groups) are not Poincare duality groups and the complexes of curves of surfaces have infinite homotopy type (i.e. are not homotopy equivalent to a finite CW-complex).

There is a natural way to associate with a transformation of an isotopy class of rational tangles to another, an element of the modular group. The correspondence between the isotopy classes of rational tangles and rational numbers follows, as well as the relation with the braid group $B_3$.

Mark and Paupert devised a general method for obtaining presentations for arithmetic non-cocompact lattices, $Γ$, in isometry groups of negatively curved symmetric spaces. The method involves a classical theorem of Macbeath applied to a $Γ$-invariant covering by horoballs of the negatively curved symmetric space upon w…

A formula for Rademacher symbols in triangle groups is provided.

Study of modular representations in homology of congruence subgroups.

Proposes a topological framework to study modular invariants and related concepts.