Modular forms Modular forms of level $1$ Modular forms of weight $2$ Dirichlet characters Eisenstein series and Bernoulli numbers Dimension formulas Linear algebra General modular symbols Computing with newforms Computing periods Solutions to selected exercises Appendix A: Computing in higher rank Bibliography Index.

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Modular forms, a computational approach

We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.

3-Manifold Groups

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.

Introduction to Arithmetic Groups

Rigidity of commensurators and irreducible lattices

We study the geometry of non-relatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with non-relatively hyperbolic peripheral subgroups is a quasi-isometry invariant. As an application, Artin groups are relatively hyperbolic if and only if freely decomposable. We also introduce a new quasi-isometry invariant of metric spaces called metrically thick, which is sufficient for a metric space to be non-hyperbolic relative to any non-trivial collection of subsets. Thick finitely generated groups include: mapping class groups of most surfaces; outer automorphism groups of most free groups; certain Artin groups; and others. Non-uniform lattices in higher rank semisimple Lie groups are thick and hence non-relatively hyperbolic, in contrast with rank one which provided the motivating examples of relatively hyperbolic groups. Mapping class groups are the first examples of non-relatively hyperbolic groups having cut points in any asymptotic cone, resolving several questions of Drutu and Sapir about the structure of relatively hyperbolic groups. Outside of group theory, Teichmuller spaces for surfaces of sufficiently large complexity are thick with respect to the Weil–Peterson metric, in contrast with Brock–Farb’s hyperbolicity result in low complexity.

Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity

We show that any two maximal disjoint unipotent subgroups of an irreducible non-cocompact lattice in a Lie group of rank atleast two generates a lattice. The proof uses techniques of the solution of the congruence subgroup problem. We show that any two maximal opposing unipotent subgroups of an irreducible lattice in a higher rank Lie Group, generate a lattice in the Lie Group. The method of proof is to use certain techniques of the solution of the congruence subgroup problem of arithmetic lattices in higher rank groups. We freely use the notation and results of [3] without giving explicit references therein. Let G be a simply connected absolutely almost simple linear algebraic group defined and isotropic over a global field K. Let C/+ be the unimpotent radical (which is defined over K) of a minimal parabolic A'-subgroup P+ of G. Let U~~ be the unipotent radical of another minimal parabolic A'-subgroup P~ of G which is opposed to P+ in the sense that U+ Γ\U~ = {1}. Let S be a finite set of places of K including all the archimedian ones, if any. We call thering A — Os — {x G K\ \x\ v < 1 for all places v of A', not in S} the ring of S-integers in K. Choose a faithful representation G <—» GLpj defined over K and define G(Os) = {g G G gij G Os, 1 < i,j < N}. The subgroups in G which are of finite index in G(Os) are called S-arithmetic groups. Define the 5-rank of G to be the sum Kv — rank(G). Given a non-zero ideal α of A and an algebraic ves

https://msp.org/pjm/1994/166-1/pjm-v166-n1-p08-p.pdf

On systems of generators of arithmetic subgroups of higher rank groups

Zariski dense subgroups of arithmetic groups

On superrigidity and arithmeticity of lattices in semisimple groups over local fields of arbitrary characteristic

We give a sufficient condition on a pair of (primitive) integral polynomials that the associated hypergeometric group (monodromy group of the corresponding hypergeometric differential equation) is an arithmetic subgroup of the integral symplectic group.

Arithmeticity of Certain Symplectic Hypergeometric Groups

We give a sufficient condition on a pair of (primitive) polynomials that the associated hypergeometric group (monodromy group of the corresponding hypergeometric differential equation) is an arithmetic subgroup of the integral symplectic group.

T. N. Venkataramana

Papers

On systems of generators of arithmetic subgroups of higher rank groups

Zariski dense subgroups of arithmetic groups

On superrigidity and arithmeticity of lattices in semisimple groups over local fields of arbitrary characteristic

Arithmeticity of Certain Symplectic Hypergeometric Groups

Arithmeticity of certain symplectic hypergeometric groups