arXiv research

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…

Study Eisenstein metrics on modular group representations.

Cone structures in quantum field theory linked to information geometry.

We prove a rigidity theorem for semi-arithmetic Fuchsian groups: If $Γ_1$, $Γ_2$ are two semi-arithmetic lattices in $\mathrm{PSL}(2,\mathbb{R})$ virtually admitting modular embeddings and $f\colonΓ_1\toΓ_2$ is a group isomorphism that respects the notion of congruence subgroups, then $f$ is induced by an inner automor…

Study of modular representations in homology of congruence subgroups.

New patterns deform Farey triangulation in symmetric space.

Study geometrically measures to decide if modular companions are conformally equivalent.

We introduce a groupoid ${\mathbf{ΠMG}}}$, called the fundamental modular groupoid, which is a variant of Penner's mapping class groupoid. We study how it relates to the surface mapping class groups and Thompson's group $\mathsf T$. We also introduce larger groupoid $\mathbf{ΩMG}$, which is related to outer automorphis…

As noticed by R.~Kulkarni, the conjugacy classes of subgroups of the modular group correspond bijectively to bipartite cuboid graphs. We'll explain how to recover the graph corresponding to a subgroup $G$ of $\mathrm{PSL}_2(\mathbb{Z})$ from the combinatorics of the right action of $\mathrm{PSL}_2(\mathbb{Z})$ on the r…

É.Ghys proved that the linking numbers of modular knots and the "missing" trefoil $K_{2,3}$ in $S^3$ coincide with the values of a highly ubiquitous function called the Rademacher symbol for ${\rm SL}_2\mathbb{Z}$. In this paper, we replace ${\rm SL}_2\mathbb{Z}=Γ_{2,3}$ by the triangle group $Γ_{p,q}$ for any coprime …

We describe the action of the automorphism group of the complex cubic x^2+y^2+z^2-xyz-2 on the homology of its fibers. This action includes the action of the mapping class group of a punctured torus on the subvarieties of its SL(2,C) character variety given by fixing the trace of the peripheral element (so-called "rela…

In this paper, we construct Laplace-Beltrami operators associated with arbitrary Riemannian metrics on noncommutative tori of any dimension. These operators enjoy the main properties of the Laplace-Beltrami operators on ordinary Riemannian manifolds. The construction takes into account the non-triviality of the group o…

New universal automorphic functions capture monstrous moonshine.

In this note, we establish a relationship between fractional Dehn twist coefficients of Riemann surface automorphisms and modular invariants of holomorphic families of algebraic curves. Specially, we give a characterization of pseudo-periodic maps with nontrivial fractional Dehn twist coefficients. We also obtain some …

We consider a smooth groupoid of the form Σ\rtimesΓwhere Σis a Riemann surface and Γa discrete pseudogroup acting on Σby local conformal diffeomorphisms. After defining a K-cycle on the crossed product C_0(Σ)\rtimesΓgeneralising the classical Dolbeault complex, we compute its Chern character in cyclic cohomology, using…

We study the structure of the invariant of K3 surfaces with involution, which we obtained using equivariant analytic torsion. It was known before that the invariant is expressed as the Petersson norm of an automorphic form on the moduli space. When the rank of the invariant sublattice of the K3-lattice with respect to …

We propose multidimensional versions of the Painlevé VI equation and its degenerations. These field theories are related to the isomonodromy problems of flat holomorphic infinite rank bundles over elliptic curves and take the form of non-autonomous Hamiltonian equations. The modular parameter of curves plays the role o…

Countable modular groups found on surfaces with infinite type.

Study knot invariants using automorphism groups of free nilpotent groups.

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.

Study shows RAAG automorphisms and outer automorphisms are not relatively hyperbolic.

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

Free groups' automorphisms have bounded orbits.

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…

Finite groups can be automorphism groups of translation surfaces with poles.

Automorphisms of pants complex are shown to be inner.

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

Study finds conditions for finite-dimensional Lie group structure in basic automorphism groups of certain Cartan foliations.

Automorphism groups of Hopf manifolds are finite and have a bounded order.

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.

Criterion for subgroup separability in outer automorphism groups.

Study growth rates of automorphisms of special groups.

The paper studies automorphisms of 2-step nilpotent Lie groups, showing continuity up to center and field automorphisms.

Study automorphisms of pure braid groups on sphere homotopy groups.

Constructs projective moduli spaces for Calabi-Yau pairs.

New proof shows smallest non-cyclic automorphism group quotients are linear 2-groups.

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…

Describes automorphism group of Rauzy diagrams.

Classifies hyperbolic manifolds with specific automorphism groups.

Study automorphism groups of quandles up to order 10.

In this paper, we investigate the structure of the automorphism groups of pure braid groups. We prove that, for $n>3$, $\Aut(P_n)$ is generated by the subgroup $\Aut_c(P_n)$ of central automorphisms of $P_n$, the subgroup $\Aut(B_n)$ of restrictions of automorphisms of $B_n$ on $P_n$ and one extra automorphism $w_n$. W…

We study manifolds endowed with an (almost) even Clifford (hermitian) structure and admitting a large automorphism group. We classify them when they are simply connected and the dimension of the automorphism group is maximal, and also prove a gap theorem for the dimension of the automorphism group.

We show that the automorphism group of the disk complex is isomorphic to the handlebody group. Using this, we prove that the outer automorphism group of the handlebody group is trivial.

Method constructs fundamental domains for Picard modular groups.

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

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

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.