arXiv subjects

Mikael Pichot

Publications and source records attributed to Mikael Pichot.

22 records · Page 2Linked to original sources

arXiv2009

The 4-string Braid group $B_4$ has property RD and exponential mesoscopic rank

We prove that the braid group $B_4$ on 4 strings, as well as its central quotient $B_4/< z>$, have the property RD of Haagerup-Jolissaint. It follows that the automorphism group $\Aut(F_2)$ of the free group $F_2$ on 2 generators has property RD. We also prove that the braid group $B_4$ is a group of intermediate rank (of dimension 3). Namely, we show that both $B_4$ and its central quotient have exponential mesoscopic rank, i.e., that they contain exponentially many large flat balls which are not included in flats.

math.GR↗

arXiv2008

Turbulence, representations, and trace-preserving actions

We establish criteria for turbulence in certain spaces of C*-algebra representations and apply this to the problem of nonclassifiability by countable structures for group actions on a standard atomless probability space (X,μ) and on the hyperfinite II_1 factor R. We also prove that the conjugacy action on the space of free actions of a countably infinite amenable group on R is turbulent, and that the conjugacy action on the space of ergodic measure-preserving flows on (X,μ) is generically turbulent.

math.OA↗

arXiv2004

Sur les espaces mesures singuliers I - Etude metrique-mesuree

Recall Jones-Schmidt theorem that an ergodic measured equivalence relation is strongly ergodic if and only if it has no nontrivial amenable quotient. We give two new characterizations of strong ergodicity, in terms of metric-measured spaces. The first one identifies strong ergodicity with the concentration property as defined, in this (foliation) setting, by Gromov \cite{Gromov00_SQ}. The second one characterize the existence of nontrivial amenable quotients in terms of "Følner sequences" in graphings naturally associated to (the leaf space of) the equivalence relation. We also present a formalization of the concept of quasi-periodicity, based on (noncommutative) measure theory. The "singular measured spaces" appearing in the title refer to the leaf spaces of measured equivalence relations.

math.DS↗

arXiv2004

Sur les espaces mesures singuliers II - Etude spectrale

We are mainly interested here in Kazhdan's property T for measured equivalence relations. Among our main results are characterizations of strong ergodicity and Kazhdan's property in terms of the spectra of diffusion operators, associated to random walks and hilbertian representations of the underlying equivalence relation. The analog spectral characterization of property T for countable groups was proved recently by Gromov \cite{Gromov03} (and Ghys \cite{Ghys03}). Our proof put together the tools developed in the group case and further crucial technical steps from the study of amenable equivalence relations in \cite{CFW81}. As an application we show how Żuk's "$λ_1 >1/2$" criterion for property T can be adapted to measured equivalence relations.

math.DS↗