Generators and relations of “lower-triangular” automorphism groups of free groups

Abstract

We consider a certain subgroup An+ of the automorphism group of a free group of rank n. It can be regarded as a free group analogue of the group Λn of integral lower-triangular matrices. We call An+ the lower-triangular automorphism group of a free group. The first aim of the paper is to give a finite presentation for An+.The abelianization of the free group induces the surjective homomorphism ρ+ from An+ to Λn. In our previous paper [18], we introduced the lower-triangular IA-automorphism group IAn+. Here we show that IAn+ coincides with the kernel of ρ+. The second aim of the paper is to give an infinite presentation for IAn+.Finally, we study a relation of the second homology groups between An+ and Λn. In particular, we compute the second homology group H2(Λn,L) by using Magnus's presentation where L is a principal ideal domain in which 2 is invertible. For example, L=Q,Z/pZ for prime p≥3. This gives a lower bound on the integral second homology group of An+.