arXiv research

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.

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…

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…

The paper proves the monodromy of a specific curve family is arithmetic.

We compute the class of arithmetic genus two Teichmueller curves in the Picard group of pseudo-Hilbert modular surfaces, distinguished according to their torsion order and spin invariant. As an application, we compute the number of genus two square-tiled surfaces with these invariants. The main technical tool is the co…

Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that e…

We study how the systole of principal congruence coverings of a Hilbert modular variety grows when the degree of the covering goes to infinity. We prove that given a Hilbert modular variety $M$ of real dimension $2n$, the sequence of principal congruence coverings $M_{I}$ eventually satisfies $$sysπ_{1}(M_{I})\geq \fra…

We give some new methods, based on Lipschitz extension theorems, for bounding filling invariants of subsets of nonpositively curved spaces. We apply our methods to find sharp bounds on higher-order Dehn functions of Sol_{2n+1}, horospheres in euclidean buildings, Hilbert modular groups, and certain S-arithmetic groups.

We calculate the Euler characteristics of all of the Teichmuller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These curves can all be embedded in Hilbert modular surfaces and our main result is that the Euler characteristic of a Teichmu…

We introduce a new fundamental domain for the cusp stabilizer of a Hilbert modular group over a real quadratic field K=Q(sqrt n). This is constructed as the union of Dirichlet domains for the maximal unipotent group, over the leaves in a foliation of the biplane. The region is the Cartesian product of the positive real…

We compute the Euler characteristics of the recently discovered series of Gothic Teichmüller curves. The main tool is the construction of 'Gothic' Hilbert modular forms vanishing at the images of these Teichmüller curves. Contrary to all previously known examples, the Euler characteristic is not proportional to the Eul…

The simplicial volume introduced by Gromov provides a topologically accessible lower bound for the minimal volume. Lafont and Schmidt proved that the simplicial volume of closed, locally symmetric spaces of non-compact type is positive. In this paper, we present a generalization of this result to certain non-compact lo…

We show that all GL(2, R)-equivariant point markings over orbit closures of primitive genus two translation surfaces arise from marking pairs of points exchanged by the hyperelliptic involution, Weierstrass points, or the golden points in the golden eigenform locus. As corollaries, we classify the holomorphically varyi…

Research classifies quadratic forms over various fields.

We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singular hermitian line bundles of finite height. In particular, the theorem applies to the log-singular metrics of Burgos-Kramer-Kühn. Our theorem is thus suitable for application to some non-compact Shimura varieties with their bundles of cusp forms…

Countable modular groups found on surfaces with infinite type.

We study the covolumes of arithmetic lattices in $PSL_2(\mathbb R)^n$ for $n\geq 2$ and identify uniform and non-uniform irreducible lattices of minimal covolume. More precisely, let $μ$ be the Euler-Poincaré measure on $PSL_2(\mathbb R)^n$ and $χ=μ/2^n$. We show that the Hilbert modular group $PSL_2(\mathfrak o_{k_{49…

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 conjecture a formula for the refined $\mathrm{SU}(3)$ Vafa-Witten invariants of any smooth surface $S$ satisfying $H_1(S,\mathbb{Z}) = 0$ and $p_g(S)>0$. The unrefined formula corrects a proposal by Labastida-Lozano and involves unexpected algebraic expressions in modular functions. We prove that our formula satisfi…

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…

We show the non-positivity of the Einstein-Hilbert action for conformal flat Riemannian metrics. The action vanishes only when the metric is constant flat. This recovers an earlier result of Fathizadeh-Khalkhali in the setting of spectral triples on noncommutative four-torus. Furthermore, computations of the gradient f…

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

Method constructs fundamental domains for Picard modular groups.

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.

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…

New structure on unitary group of Hilbert space.

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.

We first show that hypergeometric functions appear naturally as spectral functions when applying pseudo-differential calculus to decipher heat kernel asymptotic in the situation where the symbol algebra is noncommutative. Such observation leads to a unified (works for arbitrary dimension) method of computing the modula…

We give first examples of finitely generated groups having an intermediate, with values in (0,1), Hilbert space compression (which is a numerical parameter measuring the distortion required to embed a metric space into Hilbert space). These groups include certain diagram groups. In particular, we show that the Hilbert …