Ismael Morales López
I am a forth year DPhil student at the University of Oxford, working under the supervision of Dawid Kielak and Richard Wade. I am supported by the Oxford-Cocker graduate scholarship and by the Mathematical Institute.
Previously, I completed my bachelor's and master's degrees at the Universidad Autónoma de Madrid. My theses were supervised by Andrei Jaikin-Zapirain, which can be found below.
My deparmental website: https://www.maths.ox.ac.uk/people/ismael.morales
Email: morales (at) maths.ox.ac.uk
Research: I work in geometric group theory, usually in connection to L^2-invariants and profinite rigidity. I am increasingly attracted to hyperbolic spaces and CAT(0) cube complexes.
Preprints:
Prosolvable rigidity of surface groups (with Andrei Jaikin-Zapirain). arXiv
The Hanna Neumann conjecture for graphs of free groups with cyclic edge groups (with Sam Fisher). arXiv
Publications:
Virtual homology of limit groups and profinite rigidity of direct products (with Jonathan Fruchter). Israel J. Math (to appear). arXiv
Profinite properties of residually free groups. Comm. Algebra, 2024. Open access. arXiv
Parafree graphs of groups with cyclic edge groups (with Andrei Jaikin-Zapirain). J. Lond. Math. Soc., 2024. Open access. arXiv
On the profinite rigidity of free and surface groups. Math. Ann., 2024. Open access. arXiv
A finitely generated group G is termed parafree if it is residually nilpotent and there exists a free group F which has the same isomorphism types of nilpotent quotients as G. These groups were introduced by Baumslag in 1968 and many fundamental and intriguing questions about them still remain open. They have recently received much attention in connection to Remmeslenikov's conjecture on the profinite rigidity of free groups.
In this thesis we offer a new way of producing parafree groups combining techniques on pro-p groups and L^2-Betti numbers.
There are two original results. On the one hand, we can completely describe in simple terms which free products of finitely generated groups, with abelian amalgams, are parafree. Secondly, we can give a description of which abelian HNN extensions of finitely generated groups are parafree. The latter is not as explicit. Nevertheless, if G is a parafree group with two-generated abelianisation, we can describe in very concrete terms which abelian HNN extensions of G are parafree.
The aim of this exposition is to discuss the proofs of the following two results:
Helfgott's theorem on the rapid growth of generating sets of SL(2, Z/p), which uses techniques from additive combinatorics.
The influential work of Bourgain and Gamburd on expanding Cayley graphs of SL(2, Z/p) with respect to the reduction modulo p of a generic subset S of the group SL(2, Z).
These results provided many explicit families of expander graphs. We essentially reproduce their original arguments but we also implement some simplifications and improvements based on subsequent developments due to Babai, Gowers, Nikolov, Pyber and Tao. There is no claim of originality in this exposition, except in a possibly new proof of the quasirandomness of PSL(2, Z/p) that merely uses divisibility and orthogonality relations of characters.
Other publications:
Machine learning.
Neural Latent Geometry Search: Product Manifold Inference via Gromov-Hausdorff-Informed Bayesian Optimization (Haitz Sáez de Ocáriz Borde, Alvaro Arroyo, Ismael Morales, Ingmar Posner, Xiaowen Dong). Advances in Neural Information Processing Systems, 2023. Open access. arXiv
Expository.
Cuestiones existenciales en combinatoria y teoría de números: el método probabilístico. TEMat, 4, 2020. Link.
Lifting the exponent. Archimede Mathematical Journal, 6 (2), 2019. PDF.