Three papers on decision problems and profinite completions of finitely presented groups
Martin R. Bridson
Paper 1: Decision problems and profinite completions of groups
Preprint, 8 April 2008; to appear in Journal of Algebra
- This is a .pdf file
(20 pages, no figures)
We consider pairs of finitely presented, residually finite groups $P\hookrightarrow
\G$ for which the induced map of profinite completions $\hat P\to \hat\G$ is an
isomorphism. We prove that there is no algorithm that, given an arbitrary such
pair, can determine whether or not $P$ is isomorphic to $\G$.
We construct pairs for which the conjugacy problem in $\G$
can be solved in quadratic time but the conjugacy problem in $P$ is unsolvable.
Let $\mathcal J$ be the class of super-perfect groups that
have a compact classifying space and no proper subgroups of finite index.
We prove that there does not exist an algorithm that, given a
finite presentation of a group $\G$ and a guarantee that $\G\in\mathcal J$,
can determine whether or not $\G\cong\{1\}$.
We exhibit a finitely presented acyclic group $\H$ and an integer $k$
such there is no algorithm that can determine which $k$-generator
subgroups of $\H$ are perfect.
Paper 2: The Schur multiplier, profinite completions and decidability
Preprint, 7 April 2008;
to appear in the Bulletin of the London Math. Soc.
- This is a .pdf file
(5 pages, no figures)
We fix a finitely presented group $Q$ and
consider short exact sequences
$1\to N\to \G\to Q\to 1$.
The inclusion $N\hookrightarrow\G$ induces a morphism
of profinite completions $\hat N\to \hat \G$. We prove
that this is an isomorphism for
all $N$ and $\G$ if and only
if $Q$ is super-perfect and has no proper subgroups of finite index.
We prove that there is no algorithm that, given a
finitely presented,
residually finite group $\G$ and a finitely presentable
subgroup $P\hookrightarrow\G$, can determine whether or
not $\hat P\to\hat \G$ is an isomorphism.
Paper 3: Direct factors of profinite completions and decidability
Preprint, 19 March 2008; to appear in the Journal of Group Theory
- This is a .pdf file
(6 pages, no figures)
We consider finitely presented,
residually finite groups $G$ and
finitely generated normal subgroups
$A$ such that the inclusion $A\hookrightarrow G$
induces an isomorphism from the profinite completion of $A$ to a direct
factor of the profinite completion of $G$.
We explain why $A$ need not be a direct factor of a subgroup of finite index in $G$; indeed $G$
need not have a subgroup of finite index that splits as a non-trivial direct product. We prove that there
is no algorithm that can determine whether $A$
is a direct factor of a subgroup of finite index in $G$.