The Size of Г

Abstract

In this section we present a number of results which indicate that, in various respects, Г is “as large as it can be”. The points of view are the Zariski closure, the image of reduction modulo a prime, the p-adic closure, and the maximality as a discrete subgroup of G(ℝ). Note that Theorem 1.1 could be listed here - Г is “larger” than SL_d(ℤ) in that it is cocompact but the latter is not. This is paradoxical inasmuch it always takes an effort to write down elements, let alone generators of Г whereas generators of SL_d(ℤ) or other Chevalley groups are easy to find. We shall return to this theme.