arXiv subjects

Ismael Morales

Publications and source records attributed to Ismael Morales.

9 recordsLinked to original sources

arXiv2023

Prosolvable rigidity of surface groups

Surface groups are known to be the Poincar\'e Duality groups of dimension two since the work of Eckmann, Linnell and M\"uller. We prove a prosolvable analogue of this result that allows us to show that surface groups are profinitely (and prosolvably) rigid among finitely generated groups that satisfy $\mathrm{cd}(G)=2$ and $b_2^{(2)}(G)=0$. We explore two other consequences. On the one hand, we derive that if $u$ is a surface word of a finitely generated free group $F$ and $v\in F$ is measure equivalent to $u$ in all finite solvable quotients of $F$ then $u$ and $v$ belong to the same $\mathrm{Aut}(F)$-orbit. Finally, we get a partial result towards Mel'nikov's surface group conjecture. Let $F$ be a free group of rank $n\geq 3$ and let $w\in F$. Suppose that $G=F/\langle\!\langle w\rangle\!\rangle$ is a residually finite group all of whose finite-index subgroups are one-relator groups. Then $G$ is 2-free. Moreover, we show that if $H^2(G; \mathbb{Z})\neq 0$ then $G$ must be a surface group.

math.GR↗

arXiv2023

The Hanna Neumann Conjecture for graphs of free groups with cyclic edge groups

The Hanna Neumann Conjecture (HNC) for a free group $G$ predicts that $\overline{\chi}(U\cap V)\leq \overline{\chi} (U)\overline{\chi}(V)$ for all finitely generated subgroups $U$ and $V$, where $\overline{\chi}(H) = \max\{-\chi(H),0\}$ denotes the reduced Euler characteristic of $H$. A strengthened version of the HNC was proved independently by Friedman and Mineyev in 2011. Recently, Antol\'in and Jaikin-Zapirain introduced the $L^2$-Hall property and showed that if $G$ is a hyperbolic limit group that satisfies this property, then $G$ satisfies the HNC. Antol\'in and Jaikin-Zapirain established the $L^2$-Hall property for free and surface groups, which Brown and Kharlampovich extended to all limit groups. In this article, we prove the $L^2$-Hall property for graphs of free groups with cyclic edge groups that are hyperbolic relative to virtually abelian subgroups. We also give another proof of the $L^2$-Hall property for limit groups. As a corollary, we show that all these groups satisfy a strengthened version of the HNC.

math.GR↗

arXiv2023

Neural Latent Geometry Search: Product Manifold Inference via Gromov-Hausdorff-Informed Bayesian Optimization

Recent research indicates that the performance of machine learning models can be improved by aligning the geometry of the latent space with the underlying data structure. Rather than relying solely on Euclidean space, researchers have proposed using hyperbolic and spherical spaces with constant curvature, or combinations thereof, to better model the latent space and enhance model performance. However, little attention has been given to the problem of automatically identifying the optimal latent geometry for the downstream task. We mathematically define this novel formulation and coin it as neural latent geometry search (NLGS). More specifically, we introduce an initial attempt to search for a latent geometry composed of a product of constant curvature model spaces with a small number of query evaluations, under some simplifying assumptions. To accomplish this, we propose a novel notion of distance between candidate latent geometries based on the Gromov-Hausdorff distance from metric geometry. In order to compute the Gromov-Hausdorff distance, we introduce a mapping function that enables the comparison of different manifolds by embedding them in a common high-dimensional ambient space. We then design a graph search space based on the notion of smoothness between latent geometries and employ the calculated distances as an additional inductive bias. Finally, we use Bayesian optimization to search for the optimal latent geometry in a query-efficient manner. This is a general method which can be applied to search for the optimal latent geometry for a variety of models and downstream tasks. We perform experiments on synthetic and real-world datasets to identify the optimal latent geometry for multiple machine learning problems.

cs.LG↗

arXiv2023

Gromov-Hausdorff Distances for Comparing Product Manifolds of Model Spaces

Recent studies propose enhancing machine learning models by aligning the geometric characteristics of the latent space with the underlying data structure. Instead of relying solely on Euclidean space, researchers have suggested using hyperbolic and spherical spaces with constant curvature, or their combinations (known as product manifolds), to improve model performance. However, there exists no principled technique to determine the best latent product manifold signature, which refers to the choice and dimensionality of manifold components. To address this, we introduce a novel notion of distance between candidate latent geometries using the Gromov-Hausdorff distance from metric geometry. We propose using a graph search space that uses the estimated Gromov-Hausdorff distances to search for the optimal latent geometry. In this work we focus on providing a description of an algorithm to compute the Gromov-Hausdorff distance between model spaces and its computational implementation.

cs.LG↗

arXiv2023

Profinite properties of residually free groups

Henry Wilton classified when a prime three-manifold $M$ has a residually free fundamental group $\pi_1 M$. We prove that the groups $\pi_1 M\times \mathbb Z^n$ are profinitely rigid within finitely generated residually free groups. We also establish other profinite invariants of the class of residually free groups such as coherence and subgroup separability. In the course of our proofs, we generalise a lemma of Wilton and Zalesskii on profinitely recognising when a central extension of groups splits.

math.GR↗

arXiv2022

On the profinite rigidity of free and surface groups

Let $S$ be either a free group or the fundamental group of a closed hyperbolic surface. We show that if $G$ is a finitely generated residually-$p$ group with the same pro-$p$ completion as $S$, then two-generated subgroups of $G$ are free. This generalises (and gives a new proof of) the analogous result of Baumslag for parafree groups. Our argument relies on the following new ingredient: if $G$ is a residually-(torsion-free nilpotent) group and $H\leq G$ is a virtually polycyclic subgroup, then $H$ is nilpotent and the pro-$p$ topology of $G$ induces on $H$ its full pro-$p$ topology. Then we study applications to profinite rigidity. Remeslennikov conjectured that a finitely generated residually finite $G$ with profinite completion $\widehat G\cong \widehat S$ is necessarily $G\cong S$. We confirm this when $G$ belongs to a class of groups $\mathcal{H}_{ab}$ that has a finite abelian hierarchy starting with finitely generated residually free groups. This strengthens a previous result of Wilton that relies on the hyperbolicity assumption. Lastly, we prove that the group $S\times {\mathbb Z}^n$ is profinitely rigid within finitely generated residually free groups.

math.GR↗

arXiv2022

Virtual homology of limit groups and profinite rigidity of direct products

We show that the virtual second Betti number of a finitely generated, residually free group $G$ is finite if and only if $G$ is either free, free abelian or the fundamental group of a closed surface. We also prove a similar statement in higher dimensions. We then develop techniques involving rank gradients of pro-$p$ groups, which allow us to recognise direct product decompositions. Combining these ideas, we show that direct products of free and surface groups are profinitely rigid among finitely presented, residually free groups, partially resolving a conjecture of Bridson's. Other corollaries that we obtain include a confirmation of Mel'nikov's surface group conjecture in the residually free case, and a description of closed aspherical manifolds of dimension at least $5$ with a residually free fundamental group.

math.GR↗

arXiv2021

Parafree graphs of groups with cyclic edge groups

We establish a combination theorem for parafree groups. These groups were introduced by Baumslag in the sixties. One of the current motivations for a better understanding of their structure is that they show up naturally in connection with Remeslennikov's conjecture on the profinite rigidity of free groups. In this article, we determine when the fundamental group of a finite graph of groups with cyclic edge groups is parafree.

math.GR↗

arXiv2021

Embeddings into pro-p groups and the construction of parafree groups

A finitely generated group G is termed parafree if it is residually nilpotent and it has the same isomorphism types of nilpotent quotients as some free group. The two main results of this MSc. Thesis characterise the parafreeness of two constructions, namely amalgamated products with cyclic amalgams and cyclic HNN extensions.

math.GR↗