Geodesically embeds simplest arithmetic hyperbolic manifolds into higher dimensions.
problem Embedding arithmetic hyperbolic manifolds.
method Proving embedding into arithmetic hyperbolic (n+1)-manifolds or their universal mod 2 Abelian covers. result Arithmetic hyperbolic manifolds of simplest type can be geodesically embedded.
The study of systoles in arithmetic hyperbolic manifolds.
problem Understanding the systoles of arithmetic hyperbolic manifolds.
method Construction and analysis of arithmetic hyperbolic manifolds.
result Explicit bounds on volumes and systoles of arithmetic hyperbolic manifolds.
Conditions for flat manifolds as cusp cross-sections in arithmetic hyperbolic manifolds.
problem Determining when a flat manifold can be a cusp cross-section in arithmetic hyperbolic manifolds.
method Analyzing rational representations of holonomy groups and quasi-arithmetic manifolds.
result Conditions for a flat manifold to appear as a cusp cross-section in every commensurability class of arithmetic hyperbolic manifolds.
Arithmetic 3-manifolds with infinite geodesics are proven.
problem Arithmeticity of 3-manifolds with infinitely many geodesics.
method Analysis of totally geodesic surfaces in hyperbolic 3-manifolds.
result Arithmetic 3-manifolds with infinite geodesics are proven.
The paper proves non-LERFness for certain manifold groups and explores abelian amalgamations.
problem Non-LERFness of specific manifold groups.
method Study of abelian amalgamations of hyperbolic 3-manifold groups.
result Fundamental groups of certain arithmetic hyperbolic manifolds are not LERF.
The paper embeds non-arithmetic hyperbolic manifolds into higher-dimensional spaces.
problem Embedding non-arithmetic hyperbolic manifolds into higher-dimensional hyperbolic spaces.
method Using totally geodesic submanifolds and commensurability classes.
result Many non-arithmetic hyperbolic manifolds can be embedded geodesically.
The paper proves many hyperbolic manifolds have similar growth rates to all hyperbolic manifolds.
problem Understanding growth rates of hyperbolic manifolds.
method Proving existence of many compact hyperbolic manifolds as boundaries of other hyperbolic manifolds.
result Establishes that compact hyperbolic arithmetic n-manifolds have the same growth rate as all hyperbolic n-manifolds.
Classifies Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.
problem Identifying Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.
method Comprehensive classification of commensurability classes of cusped, arithmetic, and non-arithmetic complex hyperbolic 2-manifolds.
result Some Nil 3-manifolds are cross-sections in every commensurability class, while others are cross-sections in only one.
The study identifies flat manifolds with unique cusp cross-sections in arithmetic hyperbolic manifolds.
problem Characterizing flat manifolds that have unique cusp cross-sections in arithmetic hyperbolic manifolds.
method Algebraic characterization of cusp cross-sections in arithmetic hyperbolic manifolds.
result Construction of flat manifolds with unique cusp cross-sections and proof of their existence in all dimensions n≥32. Torsion in cohomology grows exponentially for certain arithmetic hyperbolic manifolds.
problem Growth of torsion in cohomology of arithmetic hyperbolic manifolds.
method Study of sequences of arithmetic subgroups of SO_0(d,1) and Spin(d,1) yielding hyperbolic manifolds.
result Torsion in cohomology grows exponentially under natural assumptions.
Paper finds new 3D shapes that can be inside a 4D space.
problem Finding new 3D shapes with specific properties.
method Examined arithmetic hyperbolic 3-manifolds and their homology.
result Discovered infinitely many 3D shapes that are rational homology spheres and can bound geometrically.
Complex hyperbolic manifolds with many totally geodesic submanifolds are arithmetic.
problem Characterizing arithmeticity of complex hyperbolic manifolds with certain submanifolds.
method Developing superrigidity theorems for complex hyperbolic lattices and proving nonexistence of certain maps.
result Finite volume complex hyperbolic n-manifolds containing infinitely many maximal totally geodesic submanifolds of dimension at least two are arithmetic. 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.
Systoles of hyperbolic manifolds are dense and related to Salem numbers.
problem Density of systoles in hyperbolic manifolds.
method Analyzing systoles and arithmetic hyperbolic manifolds.
result Systoles of closed arithmetic hyperbolic manifolds are dense in (0,+∞). Explains how arithmetic manifolds solve geometric questions about systole and kissing number.
problem Geometric questions about systole and kissing number in hyperbolic manifolds.
method Use of arithmetic manifolds to solve geometric questions.
result Answers geometric questions about systole and kissing number for dimension 2 and higher dimensions.
Gromov and Piatetski-Shapiro proved existence of finite volume non-arithmetic hyperbolic manifolds of any given dimension. In dimension four and higher, we show that there are about v^v such manifolds of volume at most v, considered up to commensurability. Since the number of arithmetic ones tends to be polynomial, alm…
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.
The paper shows that arithmetic hyperbolic 3-orbifolds have many non-commensurable pairs with similar geodesic spectra.
problem Understanding the relationship between geodesic length spectra and commensurability of arithmetic hyperbolic 3-orbifolds.
method Using a bounded gaps result for prime ideals in number fields, the paper constructs infinitely many non-commensurable pairs of arithmetic hyperbolic 3-orbifolds with similar geodesic spectra.
result Arithmetic hyperbolic 3-orbifolds can have many non-commensurable pairs with similar geodesic spectra.
New manifolds with small systoles not quasi-arithmetic.
problem Finding manifolds with small systoles not quasi-arithmetic.
method Hybrid construction of known manifolds.
result Exhibited manifolds with arbitrarily small systoles.
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.
The study finds infinite commensurability classes of hyperbolic manifolds with Salem number lengths.
problem Understanding commensurability classes of arithmetic hyperbolic manifolds.
method Analyzing Salem numbers and their relation to arithmetic hyperbolic manifolds.
result Infinitely many commensurability classes of arithmetic hyperbolic manifolds with geodesic lengths equal to logarithms of Salem numbers.
New 4D hyperbolic spaces found with minimal volume.
problem Finding minimal-volume hyperbolic 4-manifolds.
method Applying Gromov and Piatetski-Shapiro's technique to construct non-arithmetic manifolds.
result Proved existence of at least 2 commensurability classes of minimal-volume hyperbolic 4-manifolds, and constructed the smallest known non-arithmetic one.
New math shows many 3D hyperbolic shapes can fit together.
problem Counting specific 3D shapes that fit together.
method Examined both arithmetic and non-arithmetic shapes, focusing on their volume.
result The number of such shapes grows super-exponentially with volume.
New method for computing Dirichlet domains of Macfarlane manifolds.
problem Computing Dirichlet domains of hyperbolic 3-manifolds.
method Geometric interpretation of quaternion algebras, arithmetic characterization, new method for Dirichlet domains.
result Infinitely many commensurability classes of Macfarlane manifolds arise in diverse settings.
The systole of most congruence coverings of arithmetic hyperbolic manifolds is at least a certain value.
problem Estimating the systole of congruence coverings of arithmetic hyperbolic manifolds.
method Proving a lower bound on the systole of most principal congruence coverings using logarithmic and constant factors.
result The systole of most congruence coverings satisfies a specific lower bound.
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.
We prove that for n>4 there is no compact arithmetic hyperbolic n-manifold whose Euler characteristic has absolute value equal to 2. In particular, this shows the nonexistence of arithmetically defined hyperbolic rational homology n-sphere with n even different than 4.
Finite geodesic submanifolds found in certain hyperbolic manifolds.
problem Maximal geodesic submanifolds in hyperbolic hybrids.
method Structure theory of arithmetic groups, dynamics, and geometry in negative curvature.
result Finiteness of maximal geodesic submanifolds in hyperbolic hybrids.
Study shows arithmetic properties of specific hyperbolic Dehn fillings.
problem Arithmeticity of one-cusped Dehn fillings of specific link complements.
method Investigation of cusp fields, trace fields, and invariant trace fields.
result No one-cusped hyperbolic Dehn filling of the Berge manifold is arithmetic.
These are mostly expository notes based on the course of lectures on arithmetic invariants of hyperbolic manifolds given at the workshop associated with the final "Volume Conference," held at Columbia University, June 2009. Some new results are included.
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 …
We show that the conjectural cusped complex hyperbolic 2-orbifolds of minimal volume are the two smallest arithmetic complex hyperbolic 2-orbifolds. We then show that every arithmetic cusped complex hyperbolic 2-manifold of minimal volume covers one of these two orbifolds. We also give all minimal volume manifolds that…
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…
In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of tech…
The paper connects arithmetic invariants of hyperbolic 3-manifolds.
problem Understanding the arithmetic properties of hyperbolic 3-manifolds.
method Analyzes profinite completions and algebraic invariants of fundamental groups.
result Uniform lattices with isomorphic profinite completions have identical arithmetic properties.
The paper verifies hyperbolic structures on 3-manifolds using interval arithmetic.
problem Verifying hyperbolic structures on 3-manifolds using interval arithmetic.
method Extending methods by Casson, using interval arithmetic and a new theoretical result.
result Successfully verified hyperbolic structures on known examples and determined knots with hyperbolic branched covers.
Explicitly constructed 5-manifolds tessellated by prisms.
problem Constructing closed arithmetic hyperbolic 5-manifolds.
method Explicit construction and tessellation of manifolds by Coxeter simplicial prisms.
result Explicit construction of 5-manifolds with specified properties.
The paper explores cusp types in hyperbolic 4-manifolds and their commensurability classes.
problem Understanding cusp types in hyperbolic 4-manifolds and their commensurability.
method Criteria for commensurability classes containing specific cusp types.
result Infinitely many examples of commensurability classes without certain cusp types.
The study examines how non-commensurable surfaces can share length spectra.
problem Understanding how non-commensurable arithmetic hyperbolic surfaces can share length spectra.
method Investigates quantitative results on the maximum cardinality of non-commensurable surfaces sharing a fixed set of length spectra.
result Proves a number of quantitative results about the maximum cardinality of a family of pairwise non-commensurable arithmetic hyperbolic surfaces whose length spectra contain a fixed set of nonnegative real numbers.
Finite totally geodesic hypersurfaces in curved manifolds proven.
problem Characterizing totally geodesic hypersurfaces in curved manifolds.
method Analytic Riemannian manifold analysis with negative sectional curvature.
result Closed manifolds with negative curvature have only finitely many totally geodesic hypersurfaces.
In this paper we consider the cohomology of a closed arithmetic hyperbolic 3-manifold with coefficients in the local system defined by the even symmetric powers of the standard representation of SL(2,C). The cohomology is defined over the integers and is a finite abelian group. We show that the order of the 2nd cohomol…
This paper describes a general algorithm for finding the commensurator of a non-arithmetic cusped hyperbolic manifold, and for deciding when two such manifolds are commensurable. The method is based on some elementary observations regarding horosphere packings and canonical cell decompositions. For example, we use this…
New proof shows amalgamations of hyperbolic 3-manifold groups are not LERF.
problem Proving amalgamations of hyperbolic 3-manifold groups are not LERF.
method Analyzing amalgamations of fundamental groups of finite volume hyperbolic 3-manifolds.
result Amalgamations of hyperbolic 3-manifold groups are not LERF.
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.
Arithmetic Kleinian groups are distinguished by their finite quotients.
problem Distinguishing arithmetic Kleinian groups among all finitely generated residually finite groups.
method Constructing specific examples of arithmetic Kleinian groups and proving their profinite rigidity.
result Arithmetic Kleinian groups are uniquely identified by their finite quotients.
The paper studies arithmeticity and hidden symmetries in fully augmented pretzel link complements.
problem Determining arithmeticity and commensurability of fully augmented pretzel link complements.
method Careful analysis of geometry, including cusp shapes and totally geodesic surfaces.
result Construction of two infinite families of non-arithmetic fully augmented link complements.
Löbell polyhedra have small systoles and are quasi-arithmetic.
problem Finding compact hyperbolic polyhedra with small systoles.
method Elementary and conceptual means to observe systole behavior, number theoretic invariants to refine results.
result Löbell polyhedra give examples of closed hyperbolic 3-manifolds with arbitrarily small systole and are quasi-arithmetic.
The study bounds the growth of torsion in homology and counts non-commensurable hyperbolic manifolds.
problem Bounding the growth of torsion in homology of hyperbolic manifolds.
method Analyzes the growth of torsion subgroups in homology and counts non-commensurable manifolds.
result The number of non-commensurable closed hyperbolic manifolds grows exponentially with diameter.