Introduction to Arithmetic Groups

Abstract

This book provides a gentle introduction to the study of arithmetic subgroups of semisimple Lie groups. This means that the goal is to understand the group SL(n,Z) and certain of its subgroups. Among the major results discussed in the later chapters are the Mostow Rigidity Theorem, the Margulis Superrigidity Theorem, Ratner's Theorems, and the classification of arithmetic subgroups of classical groups. As background for the proofs of these theorems, the book provides primers on lattice subgroups, arithmetic groups, real rank and Q-rank, ergodic theory, unitary representations, amenability, Kazhdan's property (T), and quasi-isometries. Numerous exercises enhance the book's usefulness both as a textbook for a second-year graduate course and for self-study. In addition, notes at the end of each chapter have suggestions for further reading. (Proofs in this book often consider only an illuminating special case.) Readers are expected to have some acquaintance with Lie groups, but appendices briefly review the prerequisite background.

Approx 500 pages, several figures. Published by Deductive Press. ISBN: 978-0-9865716-0-2 (paperback); 978-0-9865716-1-9 (hardcover). A PDF file that is an exact copy of the published version can be found in the ancillary files

Cited by in corpus (70)

• Mapping class group and global Torelli theorem for hyperkahler manifolds: an erratum
• Ratner's Theorems on Unipotent Flows
• Geometric classification of 4d $\mathcal{N}=2$ SCFTs
• Special Geometry and the Swampland
• Shimura Varieties and Moduli
• Möbius disjointness for homogeneous dynamics
• Irreducible Sp-representations and subgroup distortion in the mapping class group
• Surjectivity of Galois Representations in Rational Families of Abelian Varieties
• Cofinitely Hopfian groups, open mappings and knot complements
• From the hyperbolic 24-cell to the cuboctahedron
• Maximal subgroups and von Neumann subalgebras with the Haagerup property
• Swampland geometry and the gauge couplings
• Geometric spectra and commensurability
• Stable integral simplicial volume of 3-manifolds
• Commensurating actions for groups of piecewise continuous transformations
• Holomorphic actions of higher rank lattices in dimension three
• Kakutani Equivalence of Unipotent Flows
• Convex projective manifolds with a cusp of any non-diagonalizable type
• Essential Killing fields of parabolic geometries
• On solvable compact Clifford-Klein forms
• On growth of systole along congruence coverings of Hilbert modular varieties
• Compression functions of uniform embeddings of groups into Hilbert and Banach spaces
• A tour on Hermitian symmetric manifolds
• Non left-orderability of lattices in higher rank semi-simple Lie groups
• Geometric dimension of lattices in classical simple Lie groups
• Divisible convex sets with properly embedded cones
• Beyond the Lascar Group
• Transference principles and locally symmetric spaces
• The Grothendieck-Katz Conjecture for certain locally symmetric varieties
• Equidistribution of translates of a homogeneous measure on the Borel--Serre boundary
• Hybrid bounds for the sup-norm of automorphic forms in higher rank
• Complete quaternionic Kähler manifolds with finite volume ends
• Commutators, commensurators, and ${\mathrm{PSL}}_2(\mathbb{Z})$
• Global smooth and topological rigidity of hyperbolic lattice actions
• Cohomological obstructions to Nielsen realization
• Arbitrarily large families of spaces of the same volume
• Systole inequalities for arithmetic locally symmetric spaces
• Non-vanishing cohomology classes in uniform lattices of $\text{SO}(n,\mathbb{H})$ and automorphic representations
• Periodic maximal flats are not peripheral
• L2-invariants of nonuniform lattices in semisimple Lie groups
• Type II${}_1$ factors with arbitrary countable endomorphism group
• Discrete Real Specializations of Sesquilinear Representations of the Braid Groups
• Isometric actions of quaternionic symplectic groups
• Distal higher rank lattice actions on surfaces
• Projections in Lipschitz-free spaces induced by group actions
• Bounded cohomology is not a profinite invariant
• The Chow group mod $\ell$ for a product of elliptic curves
• Connection Blocking in $SL(n,R)$ Quotients
• Higher-rank pointwise discrepancy bounds and logarithm laws for generic lattices
• Actions of automorphism groups of Lie groups
• Cusps and Commensurability Classes of Hyperbolic 4-Manifolds
• Bordifications of hyperplane arrangements and their curve complexes
• Geometry and arithmetic of semi-arithmetic Fuchsian groups
• Non-arithmetic hybrid lattices in $\mathrm{PU}(2,1)$
• Diophantine approximation on rational quadrics
• Geometric cycles and bounded cohomology for a cocompact lattice in $S{L}_n(\mathbb{R})$
• The Twisted Euclidean Algorithm: Applications to Number Theory and Geometry
• Thin subgroups isomorphic to Gromov--Piatetski-Shapiro lattices
• Entropy of group actions beyond uniform lattices
• Representations of infinite dimension orthogonal groups of quadratic forms with finite index
• Count lifts of non-maximal closed horocycles on $S{L}_N(\mathbb{Z})\backslash S{L}_N(\mathbb{R})/S{O}_N(\mathbb{R})$
• Systole on locally symmetric spaces
• Compact Lorentz manifolds with local symmetry
• Géométrie et topologie des variétés hyperboliques de grand volume
• Finiteness properties for some rational Poincaré duality groups
• Riemannian manifolds with local symmetry
• Lie groups, algebraic groups and lattices
• On the Computation of the Zariski Closure of Finitely Generated Groups of Matrices
• An embedding theorem for automorphism groups of Cartan geometries
• On The Distribution Of Angles Between Increasingly Many Short Lattice Vectors