Description du sujet de thèse

Geometric group theory beyond the world of locally compact groups


ABSTRACT. In this project, we propose to extend technics of geometric group theory developed for discrete and locally compact groups to some non-locally compact groups. The first class of groups considered is given by universal Burger-Mozes groups acting on non-locally finite regular trees. Other questions about groups acting on median continua and lattices of Polish groups are considered.

1. INTRODUCTION
Geometric group theory aims to reveal the links between the geometry of a space X andthe algebraic, combinatorial or analytic properties of a group G acting on X. In the pastdecades, it has been a very fruitful theory and it gave many results for discrete or more generally locally compact groups.

More recently, Ideas coming geometric group theory started to be used beyond the world of locally compact groups, more specifically for Polish groups (see for example [Ros23]). Letus recall that a Polish group is a topological group that is separable and metrizable with a complete metric. A key point is the fact they have the Baire property and in that sense, their topology is somehow tame.

Examples of such groups are given by homeomorphism groups of compact metrizable
spaces, isometry groups of Polish metric spaces, groups of operators of separable Banach spaces, automorphisms groups of countable structure (like graphs)...
The aim of this project of PhD thesis fits in this general objective to extend geometric
group theory for Polish groups. So the goal is to study geometry, topology and dynamics of different kinds of Polish groups.

Leaving the word of locally compact groups (which could be thought as the world of
finite dimension) make appear new unexpected phenomena. For example, in topological dynamics. By topological dynamics, we mean the study of all continuous actions of topology groups on compact spaces. For a locally compact group, there is always a free action on some compact space but for general topological groups, it is possible to have groups which always have a fixed point for any continuous action on a compact space. These groups, like the unitary group of the Hilbert space, are called extremely amenable.

In the coming sections, we describe directions that lie in this wide project.

2. UNIVERSAL BURGER-MOZES GROUPS AS POLISH GROUPS
A first class of examples to study in this PhD project is the class of Universal Burger-Mozes groups U(Γ). These groups introduced in [BM00] are constructed in the following way. Let Γ X be a permutation group of the countable (finite or infinite) space X.

Let T be the regular tree with valency |X|and Aut(T) be the automorphism groups of T.
Let E(T) be the set of edges of T, V(T) the set of vertices and let fix a coloring c : E(T) →X such that the restriction cv of c on edges attached to the vertex v is a bijection. Now the local action of g ∈Aut(T) is the bijection σ(g, x) : X →X given by cgx ◦g ◦cx. More informally,
this encodes with colors how g maps edges attached to x and to edges attached to gx.
Now, we can define the universal Burger-Mozes group U(Γ):
U(Γ) = {g ∈Aut(T), ∀v ∈V(T), σ(g, v) ∈Γ}.

Up to now, it has been essentially studied in the case Γ is finite permutation group, i.e.,
T is locally finite and U(Γ) is locally compact for the topology of pointwise convergence on vertices. Nonetheless, there is no reason to restrict ourselves to this case and actually, there is a case (where Γ is some specific example) where U(Γ) appears as the homeomorphism group of a well-known compact set.

Let f (z) = z2−1. The Julia set of such a quadratic polynomial map is the boundary of the set of points in C with bounded orbit under iterations of f . In this particular case, the Julia set is known as the Basilica Julia set and Y. Neretin proved in [Ner23] that its homeomorphism group is a group U(Γ) where Γ is the automorphism group of the separation relation associated to the dense countable cyclic order. This has been extended and studied more thoroughly in [DT25] for the Airplane Julia set and all rabbits Julia set (the Douady rabbit being a particular case for order 3).

The idea in this part of the project is to tackle the following (non exhaustive) list of questions:

(1) Decide when U(Γ) is oligomorphic for the action on V(T). Recall that a permutation
group Γ X is oligomorphic if for all n ∈N, the diagonal action Γ Xn has finitely
many orbits. These groups are well understood as topological groups. For example,
we know they have property (T).
(2) Do these groups have the automatic continuity property (any abstract homomor-
phism to a separable topological group is continuous)?
(3) Is there a unique Polish group topology?
(4) Prove that these groups do not have generic elements (group elements with a comea-
ger conjugacy class) by proving that the translation length is a continuous conjugacy
invariant.
(5) Describe the closure of conjugacy classes.
(6) Identify other concrete appearances of universal Burger-Mozes groups U(Γ) for infi-
nite Γ.
(7) Understand the topological dynamics (i.e., continuous actions on compact spaces) of
U(Γ). In particular, give a description of the universal minimal flow of these groups,
the universal minimal proximal flow and the Furstenberg boundary. The universal
minimal flow can be described in the following way: Any minimal continuous ac-
tion of a given topological group on a compact space is a quotient of the action of the
group on its universal minimal flow. So, in some sense, this universal minimal flow
contains all the topological dynamics of a given group. The universal minimal proxi-
mal flow and the Furstenberg boundary are similar objects for proximal and strongly
proximal flows (a continuous action on a compact space).

This study of U(Γ) can be thought as a follow-up of what has been done for kaleidoscopic
groups in [DMW19, Duc20, Duc23, BT23]. In particular, questions around amenability and
extreme amenability of subgroups will be crucial. Actually, the definition of kaleidoscopic
groups in [DMW19] was inspired by universal Burger-Mozes groups. These kaleidoscopic groups can be thought as equivalents of universal Burger-Mozes groups but for dendrites (roughly topological compact tree where branching points can be dense) instead of regular trees.

This first part of the project can be a very good starting point for a PhD thesis. It deals
with objects that are well known in geometric group theory but neglected in the non-locally
compact case. There is nonetheless some literature in the locally compact case [LMW17] and
more generally a large source of inspiration in the literature to tackle these questions.
This study of universal Burger-Mozes groups can be extended to almost automorphisms
of trees. Something that has never been done outside the locally compact world.

3. FUTURE DIRECTIONS
After the topological study of universal Burger-Mozes, two possibilities that fit inside the
topic of geometric group theory for Polish groups will be suggested to the PhD student.
3.1. Median continua. The first possibility can be to study what I call "Median continua"
(this has never been published). The idea is to extend what has been done for dendrites
(which are roughly compact trees) to higher dimensions.

Definition 1. A median algebra is a pair (X, m) where X is any space and m is a map X3 →X
such that
(1) For all x, y ∈X, m(x, x, y) = x,
(2) m is symmetric and
(3) m(x, y, m(z, u, v)) = m(m(x, y, z), m(x, y, u), v).
Such a map m is called a median map.

For a, b in a median algebra, the interval [a, b] is the subset {c ∈X, m(a, b, c) = c}. A subset
Y of a median algebra is convex if for all a, b ∈Y, [a, b] ⊂Y

Definition 2. A topological median algebra is a topological space X with a continuous median
m : X3 →X where X3 is endowed with the product topology.

Definition 3. A median continuum is a triple (X, m, τ) where (X, τ) is a locally convex, path-
connected continuum (metrizable connected compact space) and (X, m) is a median algebra
such that m is continuous for the product topology.

The automorphism group of a median continuum is the subgroup of homeomorphisms
of X that preserves the median map ( f (m(x, y, z)) = m( f (x), f (y), f (z)), ∀x, y, z ∈X). We
denote it Aut(X).

Here are a few examples.

Example 4. (1) Dendrons and their separable variant: dendrites
(2) CAT(0) cube complexes.
(3) L1 spaces
(4) Products of them
(5) The Hilbert cube with the majority rule.
(6) Measurable algebras over measure spaces.
(7) Asymptotic cones of separable CAT(0) cube complexes.
(8) Roller compactifications of topological median spaces with compact intervals by Fio-
ravanti.

The idea is to make a general study of these spaces and see if there are universal spaces
like Wa˙ zewski dendrites in higher dimensions.

3.2. Lattices of Polish groups. The other direction is to study lattices of Polish groups. Let
us recall that a lattice of a locally compact group G is a discrete subgroup Γ such that the
quotient space G/Γ has an invariant finite Radon measure for the left action of G. Actu-
ally, it is not had to show that if G is merely a topological group and Γ satisfies the same
properties then it implies that G has a Radon measure whose class is invariant and thus G is
locally compact. So one has to modify the definition to get a meaningful class of groups for
non-locally compact Polish groups.

Among lattices of locally compact groups, some are called uniform. This happens when
the quotient space G/Γ is compact. So we define a lattice Γ of a Polish group G to be a
discrete subgroup such that the quotient space G/Γ is coarsely bounded (we also say that Γ
is coarsely cobounded in G) in Rosendal's sense (see [Ros21, Ros23]). If G is locally compact,
one recovers the notion of uniform lattice.

A first example is given by free groups that are lattices of the automorphism group of a
regular tree of infinite valency. One can also prove that a Hilbert space (An abelian Polish
group with addition) has lattices even if it is not obvious since the subgroup of points with
integer coordinates is not cofounded since the unit cube in Rn has diagonal √n.

The first idea is to collect examples of Polish groups with lattices and groups without lat-
tices. The second idea is to identify which properties are shared by a group and its lattices.
For example, for a Polish locally group G, G has property (T) if and only if all of its lattices
have property (T). It should not be very difficult to prove that the Bergman property or the
fixed point property for Hilbert spaces (Any continuous action by isometries on a Hilbert
space has a fixed point) are shared by a group and its lattices but what about other proper-
ties? What about property (T)? The indication used in the proof for locally compact groups
is no more available for non-locally compact groups. Can one find other tools in this new
context?

REFERENCES
[BM00] Marc Burger and Shahar Mozes. Groups acting on trees: from local to global structure. Publications
Mathématiques de l'Institut des Hautes Études Scientifiques, 92(1):113–150, 2000.
[BT23] Gianluca Basso and Todor Tsankov. Topological dynamics of kaleidoscopic groups. Adv. Math.,
416:Paper No. 108915, 49, 2023.
[DMW19] Bruno Duchesne, Nicolas Monod, and Phillip Wesolek. Kaleidoscopic groups: permutation groups
constructed from dendrite homeomorphisms. Fund. Math., 247(3):229–274, 2019.
[DT25] Bruno Duchesne and Matteo Tarocchi. Homeomorphism groups of basilica, rabbit and airplane julia
sets, 2025, 2502.07762.
[Duc20] Bruno Duchesne. Topological properties of Wa ˙ zewski dendrite groups. J. Éc. polytech. Math., 7:431–
477, 2020.
[Duc23] Bruno Duchesne. A closed subgroup of the homeomorphism group of the circle with property (T).
Int. Math. Res. Not., 2023(12):10615–10640, 2023.
[LMW17] François Le Maître and Phillip Wesolek. On strongly just infinite profinite branch groups. J. Group
Theory, 20(1):1–32, 2017.
[Ner23] [Ros21] Yury A. Neretin. On the group of homeomorphisms of the basilica, 2023, 2303.11482.
Christian Rosendal. Coarse geometry of topological groups, volume 223 of Camb. Tracts Math. Cambridge:
Cambridge University Press, 2021.
[Ros23] Christian Rosendal. Geometries of topological groups. Bull. Am. Math. Soc., New Ser., 60(4):539–568,
2023.

Contexte de travail

The Institut de Mathématique d'Orsay (IMO) is a component of the University of Paris-Saclay, located on the Orsay campus. It brings together a dynamic community of researchers, faculty members, and students engaged in all areas of pure and applied mathematics. The IMO plays a central role in mathematical research both in France and internationally, in close collaboration with other prestigious institutions. It also offers high-level education, from undergraduate to doctoral studies, and is actively involved in the promotion of mathematics.

Contraintes et risques

No specific risks