Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Decision problems and profinite completions of groups   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 construct a finitely presented acyclic group $\H$ and an integer $k$ such
that there is no algorithm that can determine which $k$-generator subgroups of
$\H$ are perfect.

22
The categories of this paper are: chao-dyn
8
The categories of this paper are: hep-ex
12
The categories of this paper are: nucl-th
7
The categories of this paper are: hep-th
29
The categories of this paper are: funct-an
16
The categories of this paper are: stat
13
The categories of this paper are: quant-ph
3
The categories of this paper are: physics
28
The categories of this paper are: alg-geom
25
The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
1
The categories of this paper are: cs