Filtration (mathematics)

Outer automorphism group

Pure mathematics

Fitting subgroup

Normal subgroup

Rank (graph theory)

Automorphism

Automorphism group

Subgroup

Surface (topology)

Kernel (algebra)

Organic chemistry

p-group

Nilpotent group

Mathematics

Characteristic subgroup

Geometry

Nilpotent

Combinatorics

Chemistry

Commutator subgroup

Index of a subgroup

Central series

Group (periodic table)

Mapping class group

Duke Mathematical Journal

Let A denote either the automorphism group of the free group of rank n>=4 or the mapping class group of an orientable surface of genus n>=12 with at most 1 boundary component, and let G be either the subgroup of IA-automorphisms or the Torelli subgroup of A, respectively. For a natural number N denote by G_N the Nth term of the lower central series of G. We prove that (i) any subgroup of G containing [G,G] (in particular, the Johnson kernel in the mapping class group case) is finitely generated; (ii) if N=2 or n>=8N-4 and K is any subgroup of G containing G_N (for instance, K can be the Nth term of the Johnson filtration of G), then G/[K,K] is nilpotent and hence the abelianization of K is finitely generated; (iii) if H is any finite index subgroup of A containing G_N, with N as in (ii), then H has finite abelianization.

On finiteness properties of the Johnson filtrations