80 research outputs found
More Polish full groups
We associate to every action of a Polish group on a standard probability
space a Polish group that we call the orbit full group. For discrete groups, we
recover the well-known full groups of pmp equivalence relations equipped with
the uniform topology. However, there are many new examples, such as orbit full
groups associated to measure preserving actions of locally compact groups. In
fact, we show that such full groups are complete invariants of orbit
equivalence.
We give various characterizations of the existence of a dense conjugacy class
for orbit full groups, and we show that the ergodic ones actually have a unique
Polish group topology. Furthermore, we characterize ergodic full groups of
countable pmp equivalence relations as those admitting non-trivial continuous
character representations.Comment: Comments welcome
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
Connected Polish groups with ample generics
In this paper, we give the first examples of connected Polish groups that
have ample generics, answering a question of Kechris and Rosendal. We show that
any Polish group with ample generics embeds into a connected Polish group with
ample generics and that full groups of type III hyperfinite ergodic equivalence
relations have ample generics. We also sketch a proof of the following result:
the full group of any type III ergodic equivalence relation has topological
rank 2.Comment: New version mentioning the results Malicki obtained independently and
simultaneously http://arxiv.org/abs/1503.03919, which also answer Kechris and
Rosendal's question in a different way. Comments welcome
Bounded normal generation is not equivalent to topological bounded normal generation
We show that some derived full groups provide examples of non
simple Polish groups with the topological bounded normal generation property.
In particular, it follows that there are Polish groups with the topological
bounded normal generation property but not the bounded normal generation
property.Comment: 11 page
On full groups of non ergodic probability measure preserving equivalence relations
We give a formula relating the topological rank of the full group of an aperiodic pmp equivalence relation to the cost of its ergodic components. Furthermore, we obtain examples of full groups having a dense free subgroup whose rank is equal to the topological rank of the full group, using a Baire category argument. We then study the automatic continuity property for full groups of aperiodic equivalence relations, and find a connected metric for which they have the automatic continuity property. This allows us to give an algebraic characterization of aperiodicity for pmp equivalence relations, namely the non-existence of homomorphisms from their full groups into totally disconnected separable groups. A simple proof of the extreme amenability of full groups of hyperfinite pmp equivalence relations is also given, generalizing to the non ergodic case a result of Giordano and Pestov
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