Some subgroups of the general linear group of order two over the ring of p-adic integers

Abstract

Let p be a prime number, ℤ p the ring of p-adic integers and ℚ p its field of fractions. In this paper we are mainly concerned with the profinite structure of the general linear group GL(2, ℤ p ) and some of its subgroups. In particular, we study the subgroup SL(2, ℤ p ), the special linear group with p-adic integer entries. We determine the p-adic completion of the general linear group GL(2, ℤ)with integer entries. In contrastwith the padic completion of SL(2, ℤ) which is equal to SL(2, ℤ p ) that of GL(2, ℤ) is far to be equal to GL(2, ℤ p ). It is the semi-direct product {−1, 1} ⋉ SL(2, ℤ p ) and is a normal subgroup of GL(2, ℤ p ). It is well known that the special linear group SL(2, ℤ) is generated by two transvections. Hence SL(2, ℤ p ) is topological generated by these transvections. Moreover since ℤ p is a local ring, one can apply O. Litoff’s theorem to get short decomposition of any element of SL(2, ℤ p ). We revisit here the algebraic and profinite structures of these groups. The explicit description of these structures will be applied elsewhere to their continuous linear representations in ultrametric Banach spaces. In particular from the profinite structure of the general linear group as well as the special linear group, one can state when there exists or not a Haar measure on these groups with values in a given complete valued field. As already said, the special linear group SL(2, ℤ p ) has two topological generators, we shall prove that the general linear group GL(2, ℤ p ) admits four topological generators. As a first consequence we deduce here that any subgroup of these groups that is of finite index is open. On the other hand, the fact that these groups are finitely generated profinite groups will be applied elsewhere to their ultrametric continuous linear representations.