This talk presents the preprint https://arxiv.org/abs/1801.05801, joint work with László Márton Tóth.
Groups of automorphisms of rooted trees
Let
be a
-regular rooted tree (each vertex has
children). We assume that
is embedded in the plane, equivalently the children of each vertex are totally ordered. Let 





be the group of finitary isomorphisms of
, i.e. those automorphisms that are order-preserving on the children for all but finitely many vertices.
Such an an automorphism is prescribed by an automorphism of a finite subtree, and extended by order-preserving isomorphisms between the components of its complement. We say that it is elementary if there exists a vertex such that the automorphism permutes its children, and acts as the identity everywhere else. Such automorphisms generate 





; equivalently 





is isomorphic to an iterated wreath product of copies of the symmetric group 


.
We say that an elementary automorphism with associated permutation 




is alternating if
is alternating. The subgroup of 





generated by such automorphisms is called the finitary alternating automorphism group and denoted by 





.
In the sequel we will always assume that 

. The countable group 







then has the following property:
Any normal subgroup of
fixes a ball around the root (in other words it is the « stabiliser of a level » of
).
Invariant random subgroups
Examples
Let 


be the completion of
in 




. It is a compact group, and as such it has an invariant measure
on the boundary 
, the pushforward of the normalised Haar measure of 


, which is in fact the visual measure from the root. It follows that the stabiliser in
of a
-random point of 
is an IRS of
.
More generally, if
is a closed subset of 
we can define an invariant random subgroup in
as follows: take a random 

, and then take the (pointwise) stabiliser in
of 
. Note that this IRS can be written as an « intersection » of those obtained in the previous paragraph (as 





















). This construction yields a priori lots of distinct invariant random subgroups, as there are plenty of 


-orbits on 

.
The main result is as follows.
Theorem: Let
be a nontrivial ergodic IRS of
, and assume that almost surely
has no global fixed point on 
. Then
is induced from a finite-index subgroup of
.
For example, let 


and 


a 
-regular subtree. Then the subgroup of 





preserving 
must have a global fixed point (as its conjugacy class supports an IRS): it is possible to see by elementary means that it is in fact equal to the subgroup fixing 
pointwise.
A more descriptive result is as follows.
Theorem: Let
be a nontrivial ergodic IRS of
. There exists a closed subset
of 
such that the distribution of the fixed subset of
is the same as that of a random translate of
.
This is not a complete classification, as different IRSs of
might have the same distribution for their fixed subsets. However there is a universal construction as follows: let
be the subset given by the theorem, and 
the subtree of
such that 



. For each component
of 


choose an integer 
. Let 








be the subgroup defined as follows: it fixes 
, and the ball of radius 
around the root of
for each
. Then 











is an invariant random subgroup (where 




is a Haar-random element) and we have the following statement.
There exists 



such that
contains 











almost surely.