Quotient groups of discrete subgroups and measure theory

Abstract

~A A discrete subgroup A of a locally compact group H is called a lattice if the volume of the quotient space A \ H (with respect to Haar measure) is finite. Let F be a lattice in G. A locally compact group H is said to be amenable if there exists a left-invariant mean (see [i]) on the space of continuous complex valued functions on H, and it is nonamenable in the opposite case. The main result of this paper (Theorem 2.7) says that in the case when the rank of G is greater than i (see below for the definition of rank) and the lattice F satisfies certain conditions of nonconductivity, any normal subgroup N of the group ~ with nonamenable quotient group ~ ~/ N is finite. The main impetus for the proof of this result is the finiteness theorem for quotient groups stating that under certain conditions on G and F (in particular, when G and F satisfy the conditions of part (II) of Theorem 2.7 and F \~ G is compact) every infinite normal subgroup of F has finite index in F. In the case when the k~-ranks of all the k~-simple factors of the groups G~ are not equal to i, the finiteness theorem for quotient groups follows easily from Theorem 2.7, Khulanitskii's theorem