18 research outputs found
Bulitko's Lemma for acylindrical splittings
We generalize Bulitko's Lemma to equations over (or homomorphisms into)
groups that have -acylindrical splittings.Comment: 19 pages, 7 figures. The introduction has been modified to reflect
the fact that the result of this paper is a key component of the author's
track finding algorithm. There was also a slight change to the formulation of
the main theore
A classification of primitive permutation groups with finite stabilizers
We classify all infinite primitive permutation groups possessing a finite
point stabilizer, thus extending the seminal Aschbacher-O'Nan-Scott Theorem to
all primitive permutation groups with finite point stabilizers.Comment: Accepted in J. Algebra. Various changes, some due to the author, some
due to suggestions from readers and others due to the comments of anonymous
referee
Amenable actions, free products and a fixed point property
We investigate the class of groups admitting an action on a set with an
invariant mean. It turns out that many free products admit such an action. We
give a complete characterisation of such free products in terms of a strong
fixed point property.Comment: 12 page
On subgroups of minimal topological groups
A topological group is minimal if it does not admit a strictly coarser
Hausdorff group topology. The Roelcke uniformity (or lower uniformity) on a
topological group is the greatest lower bound of the left and right
uniformities. A group is Roelcke-precompact if it is precompact with respect to
the Roelcke uniformity. Many naturally arising non-Abelian topological groups
are Roelcke-precompact and hence have a natural compactification. We use such
compactifications to prove that some groups of isometries are minimal. In
particular, if U_1 is the Urysohn universal metric space of diameter 1, the
group Iso(U_1) of all self-isometries of U_1 is Roelcke-precompact,
topologically simple and minimal. We also show that every topological group is
a subgroup of a minimal topologically simple Roelcke-precompact group of the
form Iso(M), where M is an appropriate non-separable version of the Urysohn
space.Comment: To appear in Topology and its Applications. 39 page
Quantum symmetric Kac-Moody pairs
The present paper develops a general theory of quantum group analogs of
symmetric pairs for involutive automorphism of the second kind of symmetrizable
Kac-Moody algebras. The resulting quantum symmetric pairs are right coideal
subalgebras of quantized enveloping algebras. They give rise to triangular
decompositions, including a quantum analog of the Iwasawa decomposition, and
they can be written explicitly in terms of generators and relations. Moreover,
their centers and their specializations are determined. The constructions
follow G. Letzter's theory of quantum symmetric pairs for semisimple Lie
algebras. The main additional ingredient is the classification of involutive
automorphisms of the second kind of symmetrizable Kac-Moody algebras due to Kac
and Wang. The resulting theory comprises various classes of examples which have
previously appeared in the literature, such as q-Onsager algebras and the
twisted q-Yangians introduced by Molev, Ragoucy, and Sorba.Comment: 61 pages; some typos corrected; final versio