I am a postdoc at IMPAN (Warsaw) as part of the Maestro project "Analysis on groups" led by Piotr Nowak since October 2023.
I defended my PhD on June 2023 under the supervision of Marc Bourdon (Université de Lille) and Bertrand Rémy (ENS Lyon) at École polytechnique, funded by the Ecole doctorale de mathématiques Hadamard.
I study cohomological invariants related to the large scale geometry of locally compact groups or to measured group theory. More precisely, I study L^2 and L^p cohomology of groups and of geometric objects often related to Lie theory (such as Lie groups, buildings, Coxeter groups or Kac-Moody groups).
Email: alopez at impan.pl
Preprints
Vanishing of the second L^p-cohomology group for most semisimple groups of rank at least 3. Submitted. arXiv
We show vanishing of the second L^p-cohomology group for most higher rank semisimple groups over local fields. More precisely, we show this result for SL(4), for simple groups of rank at least 4 that are not of exceptional type or of type D_4 and for all semisimple, non-simple groups of rank at least 3. Our methods work for large values of p in the real case and for all p>1 in the non-Archimedean case. This result points towards a positive answer to Gromov's question on vanishing of L^p-cohomology of semisimple groups for all p>1 in degrees below the rank. The methods consist in using a spectral sequence à la Bourdon-Rémy, adapting a version of Mautner's phenomenon from de Cornulier-Tessera and concluding thanks to a combinatorial case-by-case study of simple Lie groups.
Top degree ℓ^p-homology and conformal dimension of buildings. Submitted. arXiv
For a non-compact finite thickness building whose Davis apartment is an orientable pseudomanifold, we compute the supremum of the set of p>1 such that its top dimensional reduced ℓ^p-cohomology is nonzero. We adapt the non-vanishing assertion of this result to any finite thickness building using the Bestvina realization. Using similar techniques, we generalize bounds obtained by Clais on the conformal dimension of some Gromov-hyperbolic buildings to any such building.
Articles
Finitely presented simple groups and measure equivalence. Colloquium Mathematicum, 172 (2023), no. 2, 261-279. arXiv
We exhibit explicit infinite families of finitely presented, Kazhdan, simple groups that are pairwise not measure equivalent. These groups are lattices acting on products of buildings. We obtain the result by studying vanishing and non-vanishing of their L^2-Betti numbers.
Teaching
2022 -2023
Tutorials L2 Analyse et convergence 1 (Université Paris-Saclay)
Tutorials L1 Algèbre 1 (Université Paris-Saclay)
Tutorials Remise à niveau analyse et algèbre, 3ème année cycle ingénieur (Polytech Paris-Saclay)
2021-2022
Tutorials L2 Analyse et convergence 1 (Université Paris-Saclay)
Tutorials L1 Algèbre 1 (Université Paris-Saclay)
2020-2021
Tutorials L2 Analyse et convergence 1 (Université Paris-Saclay)
Tutorials L2 Analyse pour physiciens (Université Paris-Saclay)