Subject: Recap: A One-Relator Group
From: ehart@center.colgate.edu
Date: Mon, 20 Nov 1995 07:14:01 -0500 (EST)

Geoff Smith has asked me to summarize the answers to my question
that was:

> What is known about the normal subgroups
> of the fundamental group of the double torus? (I am really interested
> in knowing what the homomorphisms from the group to itself can look
> like.)  The group is < a,b,c,d | a*b*a^-1*b^-1*c*d*c^-1*d^-1 >.
>

I am assuming that people don't mind using names.

Paul Robert Brown provided lots of information about ways to see
normal subgroups geometrically using covering spaces, etc.

Jim Howie's answer was brief, so here it is:

>If the image of a homomorphism f has finite index, then it is the whole
>group
>(since a proper subgroup of finite index is a surface group of higher
>genus, so needing more than 4 generators.   Since G is residually finite,
>it is Hopfian,
>so f is an isomorphism, and automorphisms of G are fairly well understood
>(mapping class groups etc.)
>
>If the image has infinite index, it is free.   But G only has free
>homomorphic images of ranks 1 and 2.   The kernel of a homomorphism onto
>a free group of
>rank 2 is the normal closure of a pair of nonparallel, nontrivial, simple
>closed paths on the surface, so (other than the fact that there are
>infinitely
>many such things), these are easy to understand.

Thanks for the answers.

Evelyn Hart ehart@colgate.edu