152 research outputs found
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 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 is unsolvable.
Let 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 such
that there is no algorithm that can determine which -generator subgroups of
\H are perfect
The strong profinite genus of a finitely presented group can be infinite
We construct the first example of a finitely-presented, residually-finite
group that contains an infinite sequence of non-isomorphic finitely-presented
subgroups such that each of the inclusion maps induces an isomorphism of
profinite completions.Comment: 10 pages, no figures. Final version to appear in Journal of the
European Math. So
The Schur multiplier, profinite completions and decidability
We fix a finitely presented group and consider short exact sequences
with finitely generated. The inclusion
induces a morphism of profinite completions . We prove that
this is an isomorphism for all and if and only if 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 and a finitely presentable subgroup ,
can determine whether or not is an isomorphism.Comment: 6 pages no figures. To appear in the Bulletin London Math So
The isomorphism problem for profinite completions of residually finite groups
We consider pairs of finitely presented, residually finite groups
. We prove that there is no algorithm that, given an
arbitrary such pair, can determine whether or not the associated map of
profinite completions is an
isomorphism. Nor do there exist algorithms that can decide whether is
surjective, or whether is isomorphic to .Comment: 12 page
Actions of automorphism groups of free groups on homology spheres and acyclic manifolds
For n at least 3, let SAut(F_n) denote the unique subgroup of index two in
the automorphism group of a free group. The standard linear action of SL(n,Z)
on R^n induces non-trivial actions of SAut(F_n) on R^n and on S^{n-1}. We prove
that SAut(F_n) admits no non-trivial actions by homeomorphisms on acyclic
manifolds or spheres of smaller dimension. Indeed, SAut(F_n) cannot act
non-trivially on any generalized Z_2-homology sphere of dimension less than
n-1, nor on any Z_2-acyclic Z_2-homology manifold of dimension less than n. It
follows that SL(n,Z) cannot act non-trivially on such spaces either. When n is
even, we obtain similar results with Z_3 coefficients.Comment: Typos corrected, reference and thanks added. Final version, to appear
in Commetarii. Math. Hel
On the difficulty of presenting finitely presentable groups
We exhibit classes of groups in which the word problem is uniformly solvable
but in which there is no algorithm that can compute finite presentations for
finitely presentable subgroups. Direct products of hyperbolic groups, groups of
integer matrices, and right-angled Coxeter groups form such classes. We discuss
related classes of groups in which there does exist an algorithm to compute
finite presentations for finitely presentable subgroups. We also construct a
finitely presented group that has a polynomial Dehn function but in which there
is no algorithm to compute the first Betti number of the finitely presentable
subgroups.Comment: Final version. To appear in GGD volume dedicated to Fritz Grunewal
Abelian covers of graphs and maps between outer automorphism groups of free groups
We explore the existence of homomorphisms between outer automorphism groups
of free groups Out(F_n) \to Out(F_m). We prove that if n > 8 is even and n \neq
m \leq 2n, or n is odd and n \neq m \leq 2n - 2, then all such homomorphisms
have finite image; in fact they factor through det: Out(F_n) \to Z/2. In
contrast, if m = r^n(n - 1) + 1 with r coprime to (n - 1), then there exists an
embedding Out(F_n) \to Out(F_m). In order to prove this last statement, we
determine when the action of Out(F_n) by homotopy equivalences on a graph of
genus n can be lifted to an action on a normal covering with abelian Galois
group.Comment: Final version, to appear in Mathematische Annalen. Minor errors and
typos corrected, including range of n in Theorem
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