arXiv subjects

Martin R Bridson

Publications and source records attributed to Martin R Bridson.

21 records · Page 2Linked to original sources

arXiv2005

Subgroups of direct products of elementarily free groups

We exploit Zlil Sela's description of the structure of groups having the same elementary theory as free groups: they and their finitely generated subgroups form a prescribed subclass E of the hyperbolic limit groups. We prove that if $G_1,...,G_n$ are in E then a subgroup $Γ\subset G_1\times...\times G_n$ is of type $\FP_n$ if and only if $Γ$ is itself, up to finite index, the direct product of at most $n$ groups from $\mathcal E$. This answers a question of Sela.

math.GR↗

arXiv2005

Normalisers in Limit Groups

Let $\G$ be a limit group, $S\subset\G$ a subgroup, and $N$ the normaliser of $S$. If $H_1(S,\mathbb Q)$ has finite $\Q$-dimension, then $S$ is finitely generated and either $N/S$ is finite or $N$ is abelian. This result has applications to the study of subdirect products of limit groups.

math.GR↗

arXiv2002

Homomorphisms from automorphism groups of free groups

The automorphism group of a finitely generated free group is the normal closure of a single element of order 2. If $m$ is less than $n$ then a homomorphism $Aut(F_n)\to Aut(F_m)$ can have cardinality at most 2. More generally, this is true of homomorphisms from $\Aut(F_n)$ to any group that does not contain an isomorphic copy of the symmetric group $S_{n+1}$. Strong constraints are also obtained on maps to groups that do not contain a copy of $W_n= (\Bbb Z/2)^n\rtimes S_n$, or of $\Bbb Z^{n-1}$. These results place constraints on how $\Aut(F_n)$ can act. For example, if $n\ge 3$ then any action of $\Aut(F_n)$ on the circle (by homeomorphisms) factors through $\text{\rm{det}}:Aut(F_n) \to \Bbb Z_2$ .

math.GR↗