34 research outputs found
Totally disconnected locally compact groups locally of finite rank
We study totally disconnected locally compact second countable (t.d.l.c.s.c.)
groups that contain a compact open subgroup with finite rank. We show such
groups that additionally admit a pro- compact open subgroup for some
finite set of primes are virtually an extension of a finite direct
product of topologically simple groups by an elementary group. This result, in
particular, applies to l.c.s.c. -adic Lie groups. We go on to prove a
decomposition result for all t.d.l.c.s.c. groups containing a compact open
subgroup with finite rank. In the course of proving these theorems, we
demonstrate independently interesting structure results for t.d.l.c.s.c. groups
with a compact open pro-nilpotent subgroup and for topologically simple
l.c.s.c. -adic Lie groups.Comment: Referee's suggestions incorporated. Main theorems for the general
locally pro-nilpotent and the general locally of finite rank cases improve
Conjugacy class conditions in locally compact second countable groups
Many non-locally compact second countable groups admit a comeagre conjugacy
class. For example, this is the case for the automorphism group of the rational
order and the automorphism group of the random graph [Truss]. A. Kechris and C.
Rosendal ask if a non-trivial locally compact second countable group can admit
a comeagre conjugacy class. We answer the question in the negative via an
analysis of locally compact second countable groups with topological conditions
on a conjugacy class.Comment: Referee's suggestions incorporate
Commensurated subgroups in finitely generated branch groups
A subgroup is commensurated if for all . We show a finitely
generated branch group is just infinite if and only if every commensurated
subgroup is either finite or of finite index. As a consequence, every
commensurated subgroup of the Grigorchuk group is either finite or finite
index.Comment: Accepted versio
Elementary totally disconnected locally compact groups
We identify the class of elementary groups: the smallest class of totally
disconnected locally compact second countable (t.d.l.c.s.c.) groups that
contains the profinite groups and the discrete groups, is closed under group
extensions of profinite groups and discrete groups, and is closed under
countable increasing unions. We show this class enjoys robust permanence
properties. In particular, it is closed under group extension, taking closed
subgroups, taking Hausdorff quotients, and inverse limits. A characterization
of elementary groups in terms of well-founded descriptive-set-theoretic trees
is then presented. We conclude with three applications. We first prove
structure results for general t.d.l.c.s.c. groups. In particular, we show a
compactly generated t.d.l.c.s.c. group decomposes into elementary groups and
topologically characteristically simple groups via group extension. We then
prove two local-to-global structure theorems: Locally solvable t.d.l.c.s.c.
groups are elementary and [A]-regular t.d.l.c.s.c. groups are elementary.Comment: Accepted version. To appear in The Proceedings of the London
Mathematical Societ
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
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