## Alessandro Carderi

Mail: alessandro.carderi  @  tu-dresden.de

Institute of geometry
Willersbau, WIL B 107
Zellescher Weg 12-14
01069 Dresden

I am a Post-Doc at TU Dresden in the Institute of Geometry. My current position is financed by the ERC Consolidator Grant No. 681207. A very short CV.
• 2015-2019 PostDoc, TU-Dresden.
• 2014-2015 ATER (attaché temporaire de recherche), ENS-Lyon.
• 2011-2015 PhD under the direction of D. Gaboriau, ENS-Lyon.
• 2010-2011 Internship with D.Gaboriau, ENS-Lyon (LLP-Erasmus).
• 2009-2011 Corso di Laurea Specialistica in Matematica (Master), Università degli Studi di Roma “La Sapienza”.
• 2006-2009 Corso di Laurea Triennale in Matematica (Bachelor), Università degli Studi di Roma “La Sapienza”.

### Research

• On Farber sequences in locally compact groups,
Preprint. arXiv. Abstract.

We prove that any sequence of lattices in a fixed locally compact group which satisfy the conclusion of the Stuck-Zimmer theorem is Farber.

• Asymptotic invariants of lattices in locally compact groups,
Preprint. arXiv. Abstract.

The aim of this work is to understand some of the asymptotic properties of sequences of lattices in a fixed locally compact group. In particular we will study the asymptotic growth of the Betti numbers of the lattices renormalized by the covolume and the rank gradient, the minimal number of generators also renormalized by the covolume. For doing so we will consider the ultraproduct of the sequence of actions of the locally compact group on the coset spaces and we will show how the properties of one of its cross sections are related to the asymptotic properties of the lattices.

• Non-standard limits of graphs and some orbit equivalence invariants (with Damien Gaboriau and Mikael de la Salle),
Submitted. arXiv. Abstract.

We consider probability measure preserving discrete groupoids, group actions and equivalence relations in the context of general probability spaces. We study for these objects the notions of cost, $\beta$-invariant and some higher-dimensional variants. We also propose various convergence results about $\ell^2$-Betti numbers and rank gradient for sequences of actions, groupoids or equivalence relations under weak finiteness assumptions. In particular we connect the combinatorial cost with the cost of the ultralimit equivalence relations. Finally a relative version of Stuck-Zimmer property is also considered.

• An exotic group as limit of finite special linear groups (with Andreas Thom),
Ann. Inst. Fourier, 68 (2018), pp. 257-273. DOI, arXiv. Abstract.

We consider the Polish group obtained as the rank-completion of an inductive limit of finite special linear groups. This Polish group is topologically simple modulo its center, it is extremely amenable and has no non-trivial strongly continuous unitary representation on a Hilbert space.

• Orbit full groups for locally compact groups (with François Le Maître),
Trans. Amer. Math. Soc., 370 (2018), pp. 2321-2349. DOI, arXiv. Abstract.

We show that the topological rank of an orbit full group generated by an ergodic, probability measure-preserving free action of a non-discrete unimodular locally compact Polish group is two. For this, we use the existence of a cross section and show that for a locally compact Polish group, the full group generated by any dense subgroup is dense in the orbit full group of the action of the group. We prove that the orbit full group of a free action of a locally compact Polish group is extremely amenable if and only if the acting group is amenable, using the fact that the full group generates the von Neumann algebra of the action.

• Ultraproducts, weak equivalence and sofic entropy ,
Preprint. arXiv. Abstract.

In this work, we study pmp actions of countable groups on arbitrary diffuse probability spaces under the point of view of weak equivalence. We will show that any such an action is weakly equivalent to an action on a standard probability space. We also propose a metric on the space of actions modulo weak equivalence which is equivalent to the topology of Abért and Elek. We will give a simpler proof of the compactness of the space, showing that convergence is characterized by ultraproducts. Using this topology, we will show that a profinite action is weakly equivalent to an ultraproduct of finite actions. Finally, combining our results with another result of Abért and Elek, we will obtain a corollary about sofic entropy. We will show that for free groups and some property (T) groups, sofic entropy of profinite actions depends crucially on the chosen sofic approximation.

• More Polish full groups (with François Le Maître),
Topol. Appl. 202 (2016), pp. 80-105. DOI, arXiv. Abstract.

We associate to every action of a Polish group on a standard probability space a Polish group that we call the \textit{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. We also 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.

• Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups (with Rémi Boutonnet),
Geom. Funct. Anal. 25 (2015), pp. 1688–1705. DOI, arXiv. Abstract.

We provide a general criterion to deduce maximal amenability of von Neumann subalgebras $L\Lambda \subset L\Gamma$ arising from amenable subgroups $\Lambda$ of discrete countable groups $\Gamma$. The criterion is expressed in terms of $\Lambda$-invariant measures on some compact $\Gamma$-space. The strategy of proof is different from S. Popa's approach to maximal amenability via central sequences, and relies on elementary computations in a crossed-product C$^*$-algebra.

• Maximal amenable subalgebras of von Neumann algebras associated with hyperbolic groups (with Rémi Boutonnet),
Math. Ann. 367 (2017), pp. 1199–1216. DOI, arXiv. Abstract.

We prove that for any infinite, maximal amenable subgroup $H$ in a hyperbolic group $G$, the von Neumann subalgebra $LH$ is maximal amenable inside $LG$. It provides many new, explicit examples of maximal amenable subalgebras in II$_1$ factors. We also prove similar maximal amenability results for direct products of relatively hyperbolic groups and orbit equivalence relations arising from measure-preserving actions of such groups.

• Ergodic theory of group actions and von Neumann algebras,
PhD thesis under the supervision of Damien Gaboriau. TEL, pdf. Abstract.

This dissertation is about measured group theory, sofic entropy and operator algebras. More precisely, we will study actions of groups on probability spaces, some fundamental properties of their sofic entropy (for countable groups), their full groups (for Polish groups) and the amenable subalgebras of von Neumann algebras associated with hyperbolic groups and lattices of Lie groups. This dissertation is composed of three parts. The first part is devoted to the study of sofic entropy of profinite actions. Sofic entropy is an invariant for actions of sofic groups defined by L. Bowen that generalize Kolmogorov's entropy. The definition of sofic entropy makes use of a fixed sofic approximation of the group. We will show that the sofic entropy of profinite actions does depend on the chosen sofic approximation for free groups and some lattices of Lie groups. The second part is based on a joint work with François Le Maître. The content of this part is based on a prepublication in which we generalize the notion of full group to probability measure preserving actions of Polish groups, and in particular, of locally compact groups. We define a Polish topology on these full groups and we study their basic topological properties, such as the topological rank and the density of aperiodic elements. The third part is based on a joint work with Rémi Boutonnet. The content of this part is based on two prepublications in which we try to understand when the von Neumann algebra of a maximal amenable subgroup of a countable group is itself maximal amenable. We solve the question for hyperbolic and relatively hyperbolic groups using techniques due to Popa. With different techniques, we will then present a dynamical criterion which allow us to answer the question for some amenable subgroups of lattices of Lie groups of higher rank.

• Cost for Discrete Measured Groupoids,
MSc thesis under the supervision of Damien Gaboriau. pdf. Abstract.

We extend the theory of cost to measured discrete groupoids. The cost of a groupoid is the infimum of the measure of its generating set. We will compute the cost of free groupoids, free products of groupoids and smooth groupoids. Given any countable group we will consider the family of actions of the group on a standard probability space and the cost of the associated groupoids. These values are included in the interval $[\cost(G),\rank(G)]$, where $\rank(G)$ is the rank of $G$ and $\cost(G)$ is the cost of the group. As in the free case, all the actions of a free group have the same cost. We will obtain that the possible costs of a free product are an interval and using free products we will construct an uncountable family of finitely generated group with the same cost values. We will show also that for direct product this set can be highly disconnected.

My MSc thesis hasn't been properly revisioned and it could contain mistakes, misprints and misquotations. pdf.

A.C. thanks François Le Maître for helping designing this webpage.