598 research outputs found
Graphs, permutations and topological groups
Various connections between the theory of permutation groups and the theory
of topological groups are described. These connections are applied in
permutation group theory and in the structure theory of topological groups.
The first draft of these notes was written for lectures at the conference
Totally disconnected groups, graphs and geometry in Blaubeuren, Germany, 2007.Comment: 39 pages (The statement of Krophollers conjecture (item 4.30) has
been corrected
Locally compact convergence groups and n-transitive actions
All sigma-compact, locally compact groups acting sharply n-transitively and
continuously on compact spaces M have been classified, except for n=2,3 when M
is infinite and disconnected. We show that no such actions exist for n=2 and
that these actions for n=3 coincide with the action of a hyperbolic group on a
space equivariantly homeomorphic to its hyperbolic boundary. We further give a
characterization of non-compact groups acting 3-properly and transitively on
infinite compact sets as non-elementary boundary transitive hyperbolic groups.
The main tool is a generalization to locally compact groups of Bowditch's
topological characterization of hyperbolic groups. Finally, in contrast to the
case n=3, we show that for n>3, if a locally compact group acts continuously,
n-properly and n-cocompactly on a locally connected metrizable compactum M,
then M has a local cut point
Cutting up graphs revisited - a short proof of Stallings' structure theorem
This is a new and short proof of the main theorem of classical structure tree
theory. Namely, we show the existence of certain automorphism-invariant
tree-decompositions of graphs based on the principle of removing finitely many
edges. This was first done in "Cutting up graphs" by M.J. Dunwoody. The main
ideas are based on the paper "Vertex cuts" by M.J. Dunwoody and the author. We
extend the theorem to a detailed combinatorial proof of J.R. Stallings' theorem
on the structure of finitely generated groups with more than one end.Comment: 12 page
Indicability, residual finiteness, and simple subquotients of groups acting on trees
We establish three independent results on groups acting on trees. The first
implies that a compactly generated locally compact group which acts
continuously on a locally finite tree with nilpotent local action and no global
fixed point is virtually indicable; that is to say, it has a finite index
subgroup which surjects onto . The second ensures that irreducible
cocompact lattices in a product of non-discrete locally compact groups such
that one of the factors acts vertex-transitively on a tree with a nilpotent
local action cannot be residually finite. This is derived from a general
result, of independent interest, on irreducible lattices in product groups. The
third implies that every non-discrete Burger-Mozes universal group of
automorphisms of a tree with an arbitrary prescribed local action admits a
compactly generated closed subgroup with a non-discrete simple quotient. As
applications, we answer a question of D. Wise by proving the non-residual
finiteness of a certain lattice in a product of two regular trees, and we
obtain a negative answer to a question of C. Reid, concerning the structure
theory of locally compact groups
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