New non-arithmetic lattice found in PU(3,1)
problem Arithmeticity of Couwenberg-Heckman-Looijenga lattices
method Study of arithmeticity and non-arithmetic lattices in PU(n,1)
result Found a non-arithmetic lattice in PU(3,1) not commensurable to Deligne-Mostow lattice
Hybrid subgroups found in non-arithmetic PU(2,1) lattices.
problem Exploring hybrid subgroups in non-arithmetic PU(2,1) lattices.
method Exploring hybrid subgroups of certain non-arithmetic lattices in PU(2,1). Showing that Mostow's lattices are virtually hybrids and some are hybrids of two non-commensurable arithmetic lattices in PU(1,1).
result Mostow's lattices are virtually hybrids and some are hybrids of two non-commensurable arithmetic lattices in PU(1,1).
Course on arithmetic lattices at EPFL.
problem Understanding arithmetic lattices.
method Introductory course on arithmetic lattices.
result Introduction to arithmetic lattices.
New property identifies arithmetic lattices from nonuniform lattices.
problem Characterizing arithmetic lattices among nonuniform lattices.
method Introduced Bounded Clustering (B-C) property.
result B-C property uniquely identifies arithmetic lattices.
We show that the non-arithmetic lattices in PO(n,1) of Belolipetsky and Thomson (2011), obtained as fundamental groups of closed hyperbolic manifolds with short systole, are quasi-arithmetic in the sense of Vinberg, and, by contrast, the well-known non-arithmetic lattices of Gromov and Piatetski-Shapiro are not quasi-a…
New thin subgroups found in special linear groups via bending techniques.
problem Finding thin subgroups of lattices in special linear groups.
method Techniques from convex projective geometry.
result Infinitely many non-commensurable lattices with thin subgroups.
New research shows certain arithmetic lattices can't be LERF.
problem Determining if arithmetic lattices are LERF.
method Analyzing trialitarian arithmetic lattices in PSO7,1(R). result Trialitarian arithmetic lattices in PSO7,1(R) are not LERF. Proves properties of arithmetic lattices and hyperbolic manifolds.
problem Properties of arithmetic lattices and hyperbolic manifolds.
method Study of normalizers of lattices and subgroup growth theory.
result Every arithmetic lattice has the property of being the normalizer of many sublattices.
Study answers arithmeticity question for normal subgroup of lattices.
problem Arithmeticity of discrete subgroups of semisimple Lie groups with dense commensurators.
method Examined normal subgroups of lattices in semisimple Lie groups.
result Positive answer to Greenberg-Shalom's question for lattices.
We prove, under the assumption of the virtual fibration conjecture for arithmetic hyperbolic 3-manifolds, that all arithmetic lattices in O(n,1), n> 4, and different from 7, are non-coherent. We also establish noncoherence of uniform arithmetic lattices of the simplest type in SU(n,1), n> 1, and of uniform lattices in …
Study C-Fuchsian subgroups of non-arithmetic lattices.
problem Understand structure and fundamental domains of C-Fuchsian subgroups. method General procedure to analyze structure and show fundamental domains lie on a complex geodesic.
result Fundamental domains of C-Fuchsian subgroups lie on a complex geodesic homeomorphic to the unit disk. The study introduces pseudo-arithmeticity for certain lattices in hyperbolic spaces.
problem Understanding the structure of certain lattices in hyperbolic spaces.
method Introducing pseudo-arithmeticity and showing covolumes relate to special values of L-functions.
result Covolumes of certain lattices correspond to rational linear combinations of special values of L-functions.
New lattices are linked to higher hypergeometric functions.
problem Understanding non-arithmetic lattices in PU(2,1).
method Showed all known non-arithmetic lattices are monodromy groups of higher hypergeometric functions.
result Non-arithmetic lattices in PU(2,1) are linked to higher hypergeometric functions.
Study of complex projective manifolds using arithmetic lattices.
problem Holomorphic convexity for toroidal compactifications of ball quotients.
method Show that Albanese mapping on an étale covering space generates jets on the interior.
result Shafarevich conjecture on holomorphic convexity satisfied in dimension 2 for arithmetic lattices.
This paper studies the covolumes of nonuniform arithmetic lattices in PU(n, 1). We determine the smallest covolume nonuniform arithmetic lattices for each n, the number of minimal covolume lattices for each n, and study the growth of the minimal covolume as n varies. In particular, there is a unique lattice (up to conj…
Computes presentations for specific arithmetic hyperbolic lattices.
problem Computing presentations for cusped arithmetic hyperbolic lattices.
method Applying Macbeath's classical result to invariant horoball covers.
result Computations for specific groups like Picard modular and quaternion hyperbolic.
Thin groups found in specific lattices.
problem Embedding right-angled Coxeter groups in arithmetic lattices.
method Using Agol's unpublished argument, embedding in indefinite orthogonal groups.
result Irreducible right-angled Coxeter groups embed as thin subgroups.
Finite actions of lattices on manifolds proven for certain groups.
problem Finite actions of lattices on compact manifolds.
method Uses machinery from Brown, Fisher, and Hurtado.
result Finite actions proven for lattices in p-adic and S-arithmetic groups. New lattices in higher dimensions have dense surface subgroups.
problem Finding dense subgroups in higher-dimensional arithmetic lattices.
method Exhibited nonuniform arithmetic lattices in SO(n,1).
result Contain Zariski-dense surface subgroups.
We show that S-arithmetic lattices in semisimple Lie groups with no rank one factors are quasi-isometrically rigid.
We describe a general procedure to produce fundamental domains for complex hyperbolic triangle groups, a class of groups that contains a representative of the commensurability class of every known non-arithmetic lattice in PU(2,1). We discuss several commensurability invariants for lattices, and show that some …
The principle result of this article is the determination of the possible finite subgroups of arithmetic lattices in U(2,1).
The paper explores subspaces in hyperbolic lattices and their arithmetic properties.
problem Arithmeticity criterion for hyperbolic lattices and suborbifolds.
method Analysis of totally geodesic suborbifolds and Vinberg's commensurability invariants.
result Arithmeticity of hyperbolic orbifolds is linked to the existence of infinitely many fc-subspaces.
The study finds that certain hyperbolic manifolds contain subgroups isomorphic to surface groups.
problem The existence of thin surface subgroups in non-uniform arithmetic lattices.
method Analyzes arithmetic hyperbolic manifolds and their fundamental groups.
result Fundamental groups of non-compact arithmetic hyperbolic manifolds contain thin surface subgroups.
Minimal crossing number found in arithmetic curve systems.
problem Finding the minimal crossing number in arithmetic curve systems.
method Analyzing systoles of hyperbolic surfaces associated with congruence lattices in SL2(Z).
result Minimal crossing number is asymptotically achieved.
New bounds on diameters and generators for specific lattices and graphs.
problem Finding bounds on diameters and generators for arithmetic lattices and Ramanujan graphs.
method Analyzing arithmetic lattices from Eichler orders in quaternion algebras, applying techniques to definite quaternion algebras.
result Bounds on diameters and generators for arithmetic lattices and Ramanujan graphs.
The paper improves Vinberg's algorithm for arithmetic hyperbolic lattices.
problem Finding maximal reflection sublattices in arithmetic hyperbolic lattices.
method Provided an effective termination condition for Vinberg's semi-algorithm.
result The algorithm becomes an effective method for finding maximal reflection sublattices.
We prove that an irreducible lattice in a semisimple algebraic group is virtually isomorphic to an arithmetic lattice if and only if it admits a faithful self-similar action on a rooted tree of finite valency.
Arithmeticity proven for certain lattices in SO(n,1) with specific geometric properties.
problem Arithmeticity of lattices in SO(n,1) with totally geodesic subspaces.
method Superrigidity theorem for certain representations of lattices, using equidistribution results from homogeneous dynamics.
result Arithmeticity of lattices proven under specific geometric conditions.
We complete the quasi-isometric classification of irreducible lattices in semisimple Lie groups over nondiscrete locally compact fields of characteristic zero by showing that any quasi-isometry of a rank one S-arithmetic lattice in a semisimple Lie group over nondiscrete locally compact fields of characteristic zero is…
We study lattices in non-positively curved metric spaces. Borel density is established in that setting as well as a form of Mostow rigidity. A converse to the flat torus theorem is provided. Geometric arithmeticity results are obtained after a detour through superrigidity and arithmeticity of abstract lattices. Residua…
We study upper bounds for the torsion in homology of nonuniform arithmetic lattices. Together with recent results of Calegari-Venkatesh, this can be used to obtain upper bounds on K2 of the ring of integers of totally imaginary fields.
We produce a family of new, non arithmetic lattices in PU(2,1). All previously known examples were commensurable with lattices constructed by Picard, Mostow and Deligne-Mostow, and fell into 9 commensurability classes. Our groups produce 5 new distinct commensurability classes. Most of the techniques are completely gen…
New lattices in higher rank contain a fixed 3-manifold group with increasing systole.
problem Finding lattices with a fixed 3-manifold group and large systole.
method Constructing arithmetic lattices in SL(8,R) with specific properties. result Existence of lattices with large systole containing a fixed 3-manifold group.
We prove there are exactly 16 arithmetic lattices of hyperbolic 3-space which are generated by two elements of finite orders p and q with p,q at least six. We also verify a conjecture of H.M. Hilden, M.T. Lozano, and J.M. Montesinos concerning the orders of the singular sets of arithmetic orbifold Dehn surgeries on two…
The paper finds many thin subgroups isomorphic to Gromov-Piatetski-Shapiro lattices.
problem Understanding thin subgroups in special linear groups.
method Constructing and embedding non-arithmetic hyperbolic manifolds into SL(n+1)(R).
result Non-arithmetic lattices in SO(n,1) can be embedded into SL(n+1)(R) as thin subgroups.
Study complex hyperbolic lattices and their subgroups, proving new finiteness properties.
problem Characterize subgroups of complex hyperbolic lattices.
method Analyzing homomorphisms and using arithmetic lattice properties.
result Deep subgroups of complex hyperbolic lattices admit homomorphisms to Z with specific kernel types.
Gopal Prasad and A. S. Rapinchuk defined a notion of weakly commensurable lattices in a semisimple group, and gave a classification of weakly commensurable Zariski dense subgroups. A motivation was to classify pairs of locally symmetric spaces isospectral with respect to the Laplacian on functions. For this, in higher …
Develops Kleinian Sphere Packings and Bugs, proving their arithmetic origins.
problem Understanding sphere packings and their arithmetic origins in various dimensions.
method Introduces Kleinian Sphere Packings and Bugs, extending Arithmeticity Theorem.
result Kleinian packings and Bugs come from Q-arithmetic lattices of simplest type.
Study jigsaw constructions of hyperbolic lattices and answer questions on arithmeticity and pseudomodularity.
problem Constructing and analyzing non-commensurable, non-uniform, non-arithmetic lattices in hyperbolic geometry.
method Hyperbolic jigsaw construction and recursive formulas for tessellations.
result Demonstration of recursive formula for tessellation of hyperbolic plane, generalizing Farey addition.
The paper studies geodesic hypersurfaces in hyperbolic manifolds and their fundamental groups.
problem Understanding the fundamental groups of hyperbolic manifolds through geodesic hypersurfaces.
method Analyzing sequences of asymptotically geodesic hypersurfaces and their properties.
result If a closed hyperbolic manifold contains a sequence of asymptotically geodesic hypersurfaces, its fundamental group is virtually special and linear over integers.
Minimal non-arithmetic hyperbolic 3-orbifold found with least volume.
problem Finding the hyperbolic 3-orbifold with minimal volume among non-arithmetic ones.
method Utilized the tetrahedral Coxeter group and horoball configuration to prove minimal volume.
result The 1-cusped quotient of hyperbolic space by the tetrahedral Coxeter group has minimal volume.
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…
This thesis investigates cusp cross-sections of arithmetic real, complex, and quaternionic hyperbolic n--orbifolds. We give a smooth classification of these submanifolds and analyze their induced geometry. One of the primary tools is a new subgroup separability result for general arithmetic lattices.
Complex hyperbolic triangle groups were first considered by Mostow in building the first nonarithmetic lattices in PU(2, 1). They are a natural generalization of the classical triangle groups acting on the hyperbolic plane. A well-known theorem of Takeuchi is that there are only finitely many Fuchsian triangle groups t…
The study counts Salem numbers linked to arithmetic hyperbolic orbifolds.
problem Bounding the proportion of Salem numbers in arithmetic lattices.
method Using results on the distribution of Salem numbers, classical methods for counting Pythagorean triples, and Gauss' lattice-counting argument.
result Improved bounds on the proportion of Salem numbers and strong exponential growth of averages.
The study finds non-uniform lattices with thin Hitchin representations in specific Lie groups.
problem Finding thin Hitchin representations in non-uniform lattices of Lie groups.
method Arithmetic methods to construct thin Hitchin representations.
result Infinitely many orbits of thin Hitchin representations in non-uniform lattices.
We prove noncoherence of certain families of lattices in the isometry group of the hyperbolic n-space for n greater than 3. For instance, every nonuniform arithmetic lattice in SO(n,1) is noncoherent, provided that n is at least 6.