The stabilizer of a column in a matrix group over a polynomial ring

Abstract

An original non-standard approach to describing the structure of a column stabilizer in a group of $n \times n$ matrices over a polynomial ring or a Laurent polynomial ring of $n$ variables is presented. The stabilizer is described as an extension of a subgroup of a rather simple structure using the $(n-1) \times (n-1)$ matrix group of congruence type over the corresponding ring of $n-1$ variables. In this paper, we consider cases where $n \leq 3.$ For $n = 2$, the stabilizer is defined as a one-parameter subgroup, and the proof is carried out by direct calculation. The case $n = 3$ is nontrivial; the approach mentioned above is applied to it. Corollaries are given to the results obtained. In particular, we prove that for the stabilizer in the question, it is not generated by its a finite subset together with the so-called tame stabilizer of the given column. We are going to study the cases when $n \geq 4$ in a forthcoming paper. Note that a number of key subgroups of the groups of automorphisms of groups are defined as column stabilizers in matrix groups. For example, this describes the subgroup IAut($M_r$) of automorphisms that are identical modulo a commutant of a free metabelian group $M_r$ of rank $r$. This approach demonstrates the parallelism of theories of groups of automorphisms of groups and matrix groups that exists for a number of well-known groups. This allows us to use the results on matrix groups to describe automorphism groups. In this work, the classical theorems of Suslin, Cohn, as well as Bachmuth and Mochizuki are used.