Sam Mellick’s research while affiliated with French National Centre for Scientific Research and other places

We characterize exactness of a countable group $\Gamma$ in terms of invariant random equivalence relations (IREs) on $\Gamma$. Specifically, we show that $\Gamma$ is exact if and only if every weak limit of finite IREs is an amenable IRE. In particular, for exact groups this implies amenability of the restricted rerooting relation associated to the ideal Bernoulli Voronoi tessellation, the discrete analog of the ideal Poisson Voronoi tesselation.

In this note, we resolve a question of D'Achille, Curien, Enriquez, Lyons, and \"Unel by showing that the cells of the ideal Poisson Voronoi tessellation are indistinguishable. This follows from an application of the Howe-Moore theorem and a theorem of Meyerovitch about the nonexistence of thinnings of the Poisson point process. We also give an alternative proof of Meyerovitch's theorem.

Let $G_1$ be a semisimple real Lie group and $G_2$ another locally compact second countable unimodular group. We prove that $G_1 \times G_2$ has fixed price one if $G_1$ has higher rank, or if $G_1$ has rank one and $G_2$ is a p-adic split reductive group of rank at least one. As an application we resolve a question of Gaboriau showing $SL(2,\mathbb{Q})$ has fixed price one. Inspired by the very recent work arXiv:2307.01194v1 [math.GT], we employ the method developed by the author and Mikl\'os Ab\'ert to show that all essentially free probability measure preserving actions of groups weakly factor onto the Cox process driven by their amenable subgroups. We then show that if an amenable subgroup can be found satisfying a double recurrence property then the Cox process driven by it has cost one.

Let G be a higher rank semisimple real Lie group or the product of at least two automorphism groups of regular trees. We prove all probability measure preserving actions of lattices in such groups have cost one, answering Gaboriau's fixed price question for this class of groups. We prove the minimal number generators of a torsion-free lattice in G is sublinear in the co-volume of $\Gamma$, settling a conjecture of Ab\'{e}rt-Gelander-Nikolov. As a consequence, we derive new estimates on the growth of first mod-p homology groups of higher rank locally symmetric spaces. Our method of proof is novel, using low intensity Poisson point processes on higher rank symmetric spaces and the geometry of their associated Voronoi tessellations. We prove as the intensities limit to zero, these tessellations partition the space into ``horoball-like'' cells so that any two share an unbounded border. We use this new phenomenon to construct low cost graphings for orbit equivalence relations of higher rank lattices.

Let G be a locally compact, second countable, unimodular group that is nondiscrete and noncompact. We explore the ergodic theory of invariant point processes on G. Our first result shows that every free probability measure preserving (pmp) action of G can be realized by an invariant point process. We then analyze the cost of pmp actions of G using this language. We show that among free pmp actions, the cost is maximal on the Poisson processes. This follows from showing that every free point process weakly factors onto any Poisson process and that the cost is monotone for weak factors, up to some restrictions. We apply this to show that G × ℤ has fixed price 1, solving a problem of Carderi. We also show that when G is a semisimple real Lie group, the rank gradient of any Farber sequence of lattices in G is dominated by the cost of the Poisson process of G. The same holds for the symmetric space X of G. This, in particular, implies that if the cost of the Poisson process of the hyperbolic 3-space ℍ3 vanishes, then the ratio of the Heegaard genus and the rank of a hyperbolic 3-manifold tends to infinity over arbitrary expander Farber sequences, in particular, the ratio can get arbitrarily large. On the other hand, if the cost of the Poisson process on ℍ3 does not vanish, it solves the cost versus L2 Betti problem of Gaboriau for countable equivalence relations.

Let G be a locally compact, second countable, unimodular group that is nondiscrete and noncompact. We explore the theory of invariant point processes on G. We show that every free probability measure preserving (pmp) action of G can be realized by an invariant point process. We analyze the cost of pmp actions of G using this language. We show that among free pmp actions, the cost is maximal on the Poisson processes. This follows from showing that every free point process weakly factors onto any Poisson process and that the cost is monotone for weak factors, up to some restrictions. We apply this to show that $G\times \mathbb{Z}$ has fixed price 1, solving a problem of Carderi. We also show that when G is a semisimple real Lie group, the rank gradient of any Farber sequence of lattices in G is dominated by the cost of the Poisson process of G. This, in particular, implies that if the cost of the Poisson process of $SL_{2}(\mathbb{C})$ vanishes, then the ratio of the Heegaard genus and the rank of a hyperbolic 3-manifold tends to infinity over Farber chains.

We define a probability measure preserving and r-discrete groupoid that is associated to every invariant point process on a locally compact and second countable group. This groupoid governs certain factor processes of the point process, in particular the existence of Cayley factor graphs. With this method we are able to show that point processes on amenable groups admit all (and only admit) Cayley factor graphs of amenable groups, and that the Poisson point process on groups with Kazhdan's Property (T) admits no Cayley factor graphs. This gives examples of pmp countable Borel equivalence relations that cannot be generated by any free action of a countable group.