52 research outputs found
Decision problems for 3-manifolds and their fundamental groups
We survey the status of some decision problems for 3-manifolds and their
fundamental groups. This includes the classical decision problems for finitely
presented groups (Word Problem, Conjugacy Problem, Isomorphism Problem), and
also the Homeomorphism Problem for 3-manifolds and the Membership Problem for
3-manifold groups.Comment: 31 pages, final versio
Braided surfaces and their characteristic maps
We show that branched coverings of surfaces of large enough genus arise as
characteristic maps of braided surfaces, thus being -prems. In the reverse
direction we show that any nonabelian surface group has infinitely many finite
simple nonabelian groups quotients with characteristic kernels which do not
contain any simple loops and hence the quotient maps do not factor through free
groups. By a pullback construction, finite dimensional Hermitian
representations of braid groups provide invariants for the braided surfaces. We
show that the strong equivalence classes of braided surfaces are separated by
such invariants if and only if they are profinitely separated.Comment: 20
Finitely presented wreath products and double coset decompositions
We characterize which permutational wreath products W^(X)\rtimes G are
finitely presented. This occurs if and only if G and W are finitely presented,
G acts on X with finitely generated stabilizers, and with finitely many orbits
on the cartesian square X^2. On the one hand, this extends a result of G.
Baumslag about standard wreath products; on the other hand, this provides
nontrivial examples of finitely presented groups. For instance, we obtain two
quasi-isometric finitely presented groups, one of which is torsion-free and the
other has an infinite torsion subgroup.
Motivated by the characterization above, we discuss the following question:
which finitely generated groups can have a finitely generated subgroup with
finitely many double cosets? The discussion involves properties related to the
structure of maximal subgroups, and to the profinite topology.Comment: 21 pages; no figure. To appear in Geom. Dedicat
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