14,805 research outputs found
On strongly just infinite profinite branch groups
For profinite branch groups, we first demonstrate the equivalence of the
Bergman property, uncountable cofinality, Cayley boundedness, the countable
index property, and the condition that every non-trivial normal subgroup is
open; compact groups enjoying the last condition are called strongly just
infinite. For strongly just infinite profinite branch groups with mild
additional assumptions, we verify the invariant automatic continuity property
and the locally compact automatic continuity property. Examples are then
presented, including the profinite completion of the first Grigorchuk group. As
an application, we show that many Burger-Mozes universal simple groups enjoy
several automatic continuity properties.Comment: Typos and a minor error correcte
On conjugacy separability of fibre products
In this paper we study conjugacy separability of subdirect products of two
free (or hyperbolic) groups. We establish necessary and sufficient criteria and
apply them to fibre products to produce a finitely presented group in
which all finite index subgroups are conjugacy separable, but which has an
index overgroup that is not conjugacy separable. Conversely, we construct a
finitely presented group which has a non-conjugacy separable subgroup of
index such that every finite index normal overgroup of is conjugacy
separable. The normality of the overgroup is essential in the last example, as
such a group will always posses an index overgroup that is not
conjugacy separable.
Finally, we characterize -conjugacy separable subdirect products of two
free groups, where is a prime. We show that fibre products provide a
natural correspondence between residually finite -groups and -conjugacy
separable subdirect products of two non-abelian free groups. As a consequence,
we deduce that the open question about the existence of an infinite finitely
presented residually finite -group is equivalent to the question about the
existence of a finitely generated -conjugacy separable full subdirect
product of infinite index in the direct product of two free groups.Comment: v2: 38 pages; this is the version accepted for publicatio
Equivariant properties of symmetric products
The filtration on the infinite symmetric product of spheres by the number of
factors provides a sequence of spectra between the sphere spectrum and the
integral Eilenberg-Mac Lane spectrum. This filtration has received a lot of
attention and the subquotients are interesting stable homotopy types. While the
symmetric product filtration has been a major focus of research since the
1980s, essentially nothing was known when one adds group actions into the
picture.
We investigate the equivariant stable homotopy types, for compact Lie groups,
obtained from this filtration of infinite symmetric products of representation
spheres. The situation differs from the non-equivariant case, for example the
subquotients of the filtration are no longer rationally trivial and on the
zeroth equivariant homotopy groups an interesting filtration of the
augmentation ideals of the Burnside rings arises. Our method is by global
homotopy theory, i.e., we study the simultaneous behavior for all compact Lie
groups at once.Comment: 33 page
On semilinear representations of the infinite symmetric group
In this note the smooth (i.e. with open stabilizers) linear and {\sl
semilinear} representations of certain permutation groups (such as infinite
symmetric group or automorphism group of an infinite-dimensional vector space
over a finite field) are studied. Many results here are well-known to the
experts, at least in the case of {\sl linear representations} of symmetric
group. The presented results suggest, in particular, that an analogue of
Hilbert's Theorem 90 should hold: in the case of faithful action of the group
on the base field the irreducible smooth semilinear representations are
one-dimensional (and trivial in appropriate sense).Comment: 19 pages, significant changes; an analogue of Hilbert's Theorem 90
for infinite symmetric groups moved to arXiv:1508.0226
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
On sequences of finitely generated discrete groups
We consider sequences of finitely generated discrete subgroups
Gamma_i=rho_i(Gamma) of a rank 1 Lie group G, where the representations rho_i
are not necessarily faithful. We show that, for algebraically convergent
sequences (Gamma_i), unless Gamma_i's are (eventually) elementary or contain
normal finite subgroups of arbitrarily high order, their algebraic limit is a
discrete nonelementary subgroup of G. In the case of divergent sequences
(Gamma_i) we show that the limiting action on a real tree T satisfies certain
semistability condition, which generalizes the notion of stability introduced
by Rips. We then verify that the group Gamma splits as an amalgam or HNN
extension of finitely generated groups, so that the edge group has an amenable
image in the isometry group of T.Comment: 21 pages, 1 figur
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