arXiv research

Study on curves on specific arithmetic quotients of hyperbolic 2-ball.

Develops arithmetic PDE geometry using Fermat quotients.

New method constructs non-arithmetic hyperbolic orbifolds from complex arithmetic ball quotients.

The paper extends arithmetic quotient results to right-angled Artin groups.

We apply G. Prasad's volume formula for the arithmetic quotients of semi-simple groups and Bruhat-Tits theory to study the covolumes of arithmetic subgroups of SO(1,n). As a result we prove that for any even dimension n there exists a unique compact arithmetic hyperbolic n-orbifold of the smallest volume. We give a for…

We construct some non-arithmetic ball quotients as branched covers of a quotient of an Abelian surface by a finite group, and compare them with lattices that previously appear in the literature. This gives an alternative construction, which is independent of the computer, of some lattices constructed by the author with…

To every $Q$-irreducible representation $r$ of a finite group $H$, there corresponds a simple factor $A$ of $Q[H]$ with an involution $τ$. To this pair $(A,τ)$, we associate an arithmetic group $Ω$ consisting of all $(2g-2)\times (2g-2)$ matrices over a natural order of $A^{op}$ which preserve a natural skew-Hermitian …

Study shows nontrivial intersections of subgroups on homogeneous spaces.

Polynomial density theorem for specific subgroup orbits in quotient spaces.

We determine the minimal volume of arithmetic hyperbolic orientable n-dimensional orbifolds (compact and non-compact) for every odd dimension n>3. Combined with the previously known results it solves the minimal volume problem for arithmetic hyperbolic n-orbifolds in all dimensions.

Define an arithmetic variety to be the quotient of a bounded symmetric domain by an arithmetic group. An arithmetic variety is algebraic, and the theorem in question states that when one applies an automorphism of the field of complex numbers to the coefficients of an arithmetic variety the resulting variety is again a…

Using the Arthur-Selberg trace formula we express the index of a Dirac operator on an arithmetic quotient over a totally real field with at least two real embeddings as the integral over the index form plus a sum of orbital integrals. For the Euler operator these orbital integrals are shown to vanish for products of ra…

Polynomial error equidistribution for SL2 groups.

There are errors in the proof of the uniqueness of arithmetic subgroups of the smallest covolume. In this note we correct the proof, obtain certain results which were stated as a conjecture, and we give several remarks on further developments.

We show that up to commensurability there are only finitely many cocompact arithmetic Kleinian groups generated by rotations. This implies, in particular, that there exist only finitely many conjugacy classes of cocompact two generated arithmetic Kleinian groups. The proof of the main result is based on a generalized G…

We prove that the uniformizing map of any arithmetic quotient, as well as the period map associated to any pure polarized $\mathbb{Z}$-variation of Hodge structure $\mathbb{V}$ on a smooth complex quasi-projective variety $S$, are topologically tame. As an easy corollary of these results and of Peterzil-Starchenko's o-…

We determine the Lyapunov spectrum of ball quotients arising from cyclic coverings. The computations are performed by rewriting the sum of Lyapunov exponents as ratios of intersection numbers and by the analysis of the period map near boundary divisors. As a corollary, we complete the classification of commensurability…

We give an algebro-geometric construction of some of the non-arithmetic ball quotients constructed by the author, Parker and Paupert. The new construction reveals a relationship between the corresponding orbifold fundamental groups and the automorphism group of the Klein quartic, and also with groups constructed by Bar…

Minimal non-arithmetic hyperbolic 3-orbifold found with least volume.

We study Deraux's non arithmetic orbifold ball quotient surfaces obtained as birational transformations of a quotient $X$ of a particular Abelian surface $A$. Using the fact that $A$ is the Jacobian of the Bolza genus $2$ curve, we identify $X$ as the weighted projective plane $\mathbb{P}(1,3,8)$. We compute the equati…

We consider the analogue of Hurwitz curves, smooth projective curves $C$ of genus $g \ge 2$ that realize equality in the Hurwitz bound $|\mathrm{Aut}(C)| \le 84 (g - 1)$, to smooth compact quotients $S$ of the unit ball in $\mathbb{C}^2$. When $S$ is arithmetic, we show that $|\mathrm{Aut}(S)| \le 288 e(S)$, where $e(S…

Paper finds new ball quotients from curve products.

We classify the minimum volume smooth complex hyperbolic surfaces that admit smooth toroidal compactifications, and we explicitly construct their compactifications. There are five such surfaces and they are all arithmetic, i.e., they are associated with quotients of the ball by an arithmetic lattice. Moreover, the asso…

The paper proves nonvanishing cohomology for ball quotient fundamental groups.

Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The ``quantization commutes with reduction'' theorem asserts that the G-invariant part of the equivariant index of M is equal to the Riemann-Roch number of the symplectic quotient of M, provided the quo…

We study fundamental groups of toroidal compactifications of non compact ball quotients and show that the Shafarevich conjecture on holomorphic convexity for these complex projective manifolds is satisfied in dimension 2 provided the corresponding lattice is arithmetic and small enough. The method is to show that the A…

The paper connects arithmetic invariants of hyperbolic 3-manifolds.

Let G be a Chevalley group scheme and B<=G a Borel subgroup scheme, both defined over Z. Let K be a global function field, S be a finite non-empty set of places over K, and O_S be the corresponding S-arithmetic ring. Then, the S-arithmetic group B(O_S) is of type F_{|S|-1} but not of type FP_{|S|}. Moreover one can der…

New bounds on diameters and generators for specific lattices and graphs.

Study on curvature properties and Shafarevich conjecture for complex hyperbolic manifolds.

This study examines arithmetic properties of GIB manifolds and their monodromy representations.

Let $M = \mathbb{B}^2 / Γ$ be a smooth ball quotient of finite volume with first betti number $b_1(M)$ and let $\mathcal{E}(M) \ge 0$ be the number of cusps (i.e., topological ends) of $M$. We study the growth rates that are possible in towers of finite-sheeted coverings of $M$. In particular, $b_1$ and $\mathcal{E}$ h…

In this paper we prove the following results: $1)$ We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important example is given by Hodge varieties, which parametrize pure polarized integral Ho…

This paper gives a complete parametrization of the commensurability classes of totally geodesic subspaces of irreducible arithmetic quotients of $X_{a, b} = (\mathbf{H}^2)^a \times (\mathbf{H}^3)^b$. A special case describes all Shimura subvarieties of type $\mathrm{A}_1$ Shimura varieties. We produce, for any $n\geq 1…

We attack a conjecture of J. Rogawski: any cocompact lattice in $S U (2, 1)$ for which the ball quotient $X = B^2 / Γ$ satisfies $b_1 (X) = 0$ and $H^{1, 1} (X) \cap H^2 (X, \bbq) \approx \bbq$ is arithmetic. We prove the Archimedian suprerigidity for representation of $Γ$ is $S L (3, \bbc)$.

A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. We compute a mapping class group of a hypekahler manifold $M$, showing that it is commensurable to an arithmetic subgroup in SO(3, b_2-3). A Teichmuller space of $M$ is a space of complex structures on $M$ up to is…

Arithmetic topology connects surface and $p$-adic field studies, enabling new insights into Galois groups.

This paper is about cohomology of mapping class groups from the perspective of arithmetic groups. For a closed surface $S$ of genus $g$, the mapping class group $Mod(S)$ admits a well-known arithmetic quotient $Mod(S)\rightarrow Sp(2g, Z)$, under which the stable cohomology of $Sp(2g,Z)$ pulls back to algebra generated…

Building on results of Arthur and Mok, we extend to (finite volume) complex and quaternionic hyperbolic manifolds the results of arXiv:1004.1085. For the spherical spectrum our results are optimal. Finally, as an application we prove a Lefschetz property for the restriction map between arithmetic quotients of complex b…

In the present article, we provide examples of fake quadrics, that is, minimal complex surfaces of general type with the same numerical invariants as the smooth quadric in $\PP ^3$ which are quotients of the bidisc by an irreducible lattice of automorphisms. Moreover, we list classes of arithmetic lattices over a real …

We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group $\mathrm{PSL}(2,\mathbb{Z}[ω])$ with $ω^2+ω+1=0$ is rigid in this sense. Other examples include th…

Let $G$ be a higher-rank semisimple Lie group over a nonarchimedean local field, for example $G={\rm PGL}(n,Q_P)$. To any lattice $L$ in $G$ there is an associated simplicial complex $B_L$, given by the quotient by $L$ of the Bruhat-Tits building associated to $G$. In this paper prove that the simplicial structure $B_L…

Proves polynomial error rate for equidistribution of unipotent flows.

We give an affirmative answer to many cases of a question due to Shalom, which asks if the commensurator of a thin subgroup of a Lie group is discrete. In this paper, let $K<Γ<G$ be an infinite normal subgroup of an arithmetic lattice $Γ$ in a rank one simple Lie group $G$, such that the quotient $Q=Γ/K$ is infinite. W…

Study on scattering geodesics on modular surface and their sojourn times.

Let $\C(Γ)$ be the set of isomorphism classes of the finite groups that are homomorphic images of $Γ$. We investigate the extent to which $\C(Γ)$ determines $Γ$ when $Γ$ is a group of geometric interest. If $Γ_1$ is a lattice in ${\rm{PSL}}(2,\R)$ and $Γ_2$ is a lattice in any connected Lie group, then $\C(Γ_1) = \C(Γ_…

Study on moduli spaces of sextic curves with simple singularities and their compactifications.

New computations show symplectic groups and mapping class groups have different properties regarding torsion.