M. S. Raghunathan

The word and Riemannian metrics on lattices of semisimple groups

Let G be a semisimple Lie group of rank ⩾2 and Γ an irreducible lattice. Γ has two natural metrics: a metric inherited from a Riemannian metric on the ambient Lie group and a word metric defined with respect to some finite set of generators. Confirming a conjecture of D. Kazhdan (cf. Gromov [Gr2]) we show that these metrics are Lipschitz equivalent. It is shown that a cyclic subgroup of Γ is virtually unipotent if and only if it has exponential growth with respect to the generators of Γ.

A Note on Quotients of Real Algebraic Groups by Arithmetic Subgroups

Introduction Let G be a connected semi-simple algebraic group defined over Q. Let F be an arithmetic subgroup of G, i.e., a subgroup of G such that for some (and therefore any) faithful rational representation p: G--, GL(N, C) defined over Q, Fc~p-I(SL(N, Z)) is of finite index in both F and p-a(SL(N, Z)). Let K c G be a maximal compact subgroup of Ga, the set of real points of G. With this notation, we can state the main result of this note.

The congruence subgroup problem

This is a short survey of the progress on the congruence subgroup problem since the sixties when the first major results on the integral unimodular groups appeared. It is aimed at the non-specialists and avoids technical details.

On the congruence subgroup problem: determination of the "Metaplectic Kernel"

Let F be a global field (i.e. either a number field or a function field in one variable over a finite field) and let oo be the set of its archimedean places. Let G be a finite non-empty set of places of F containing oo. Let o = o ( ~ ) denote the ring of G-integers of F. Let A(~) denote the ring of G-adeles i.e. the restricted direct product of the completions F v for r eG. Let f# be an absolutely simple*, simply connected subgroup of SL, defined over F. Recall that a subgroup F of C#(F) is an G-arithmetic subgroup if FnSL(n, o) has finite index in F as well as in ~(o):=fY(F)c~SL(n,o). An G-arithmetic subgroup F is a ~congruence subgroup if there exists a non-zero ideal a in o (= o(G)) such that