arXiv subjects

Eduardo Silva

Publications and source records attributed to Eduardo Silva.

12 recordsLinked to original sources

arXiv2026

Free Burnside groups of large odd exponent have cost 1

We prove that for every non-elementary torsion-free hyperbolic group $H$ and sufficiently large odd $n>1$, the quotient $H/H^n$ has cost $1$. In particular, when $H$ is the free group of rank $m\ge 2$, we obtain that the free Burnside group $B(m,n)$ has cost $1$ for every odd exponent $n\geq 665$. It follows that $B(m,n)$ is anti-treeable, and that its first $\ell^2$-Betti number $β_1^{(2)}(B(m,n))$ vanishes. The latter recovers and extends a result of Feldkamp and Kionke [Proc. Amer. Math. Soc., 2023], who proved that $β_1^{(2)}(B(m,p))=0$ for all sufficiently large prime exponents $p$.

math.GR↗

arXiv2026

Stationary boundaries on the space of amenable subgroups and C*-simplicity

We give a sufficient condition for a countable group $G$ to possess a probability measure $μ$ that admits a non-trivial $μ$-boundary modeled in the space $\mathrm{Sub}_{\mathrm{am}}(G)$ of amenable subgroups of $G$. In particular, for such $μ$ the space $\mathrm{Sub}_{\mathrm{am}}(G)$ is not uniquely $μ$-stationary. This contrasts with a theorem of Hartman-Kalantar, which states that a countable group $G$ is C*-simple if and only if there exists $μ\in \mathrm{Prob}(G)$ such that $\mathrm{Sub}_{\mathrm{am}}(G)$ is uniquely $μ$-stationary. Our criterion applies to (permutational) wreath products, which include groups that are C*-simple, and to Thompson's group $F$, whose C*-simplicity is equivalent to its non-amenability and therefore remains an open problem. We also show that any non-trivial $μ$-boundary modeled on $\mathrm{Sub}_{\mathrm{am}}(G)$ is supported on amenable normalish subgroups, in the sense of Breuillard-Kalantar-Kennedy-Ozawa. As a consequence, we conclude that a countable group with no finite normal subgroups and no amenable normalish subgroups acts essentially freely on all its Poisson boundaries.

math.GR↗

arXiv2026

Continuity of asymptotic entropy on free solvable groups

We prove the continuity of asymptotic entropy, as a function of the step distribution, among non-degenerate probability measures with finite Shannon entropy on the free solvable group $S_{d,m}$ of rank $d\ge 3$ and derived length $m\ge 2$.

math.GR↗

arXiv2026

The Poisson boundary of wreath products

We give a complete description of the Poisson boundary of wreath products $A\wr B= \bigoplus_{B} A\rtimes B$ of countable groups $A$ and $B$, for probability measures $μ$ with finite entropy where lamp configurations stabilize almost surely. If, in addition, the projection of $μ$ to $B$ is Liouville, we prove that the Poisson boundary of $(A\wr B,μ)$ is equal to the space of limit lamp configurations, endowed with the corresponding hitting measure. In particular, this answers an open question asked by Kaimanovich, and Lyons-Peres, for $B=\mathbb{Z}^d$, $d\ge 3$, and measures $μ$ with a finite first moment.

math.GR↗

arXiv2026

Continuity of asymptotic entropy on wreath products

We prove the continuity of asymptotic entropy as a function of the step distribution for non-degenerate probability measures with finite entropy on wreath products $ A \wr B = \bigoplus_B A \rtimes B $, where $A$ is any countable group and $B$ is a countable hyper-FC-central group that contains a finitely generated subgroup of at least cubic growth. As one step in proving the above, we show that on any countable group $G$ the probability that the $μ$-random walk on $G$ never returns to the identity is continuous in $μ$, for measures $μ$ such that the semigroup generated by the support of $μ$ contains a finitely generated subgroup of at least cubic growth. Finally, we show that among random walks on a group $G$ that admit a separable completely metrizable space $X$ as a model for their Poisson boundary, the weak continuity of the associated harmonic measures on $X$ implies the continuity of the asymptotic entropy. This result recovers the continuity of asymptotic entropy on known cases, such as Gromov hyperbolic groups and acylindrically hyperbolic groups, and extends it to new classes of groups, including linear groups and groups acting on $\mathrm{CAT}(0)$ spaces.

math.GR↗

arXiv2025

Bayesian Optimization and Convolutional Neural Networks for Zernike-Based Wavefront Correction in High Harmonic Generation

High harmonic generation (HHG) is a nonlinear process that enables table-top generation of tunable, high-energy, coherent, ultrashort radiation pulses in the extreme ultraviolet (EUV) to soft X-ray range. These pulses find applications in photoemission spectroscopy in condensed matter physics, pump-probe spectroscopy for high-energy-density plasmas, and attosecond science. However, optical aberrations in the high-power laser systems required for HHG degrade beam quality and reduce efficiency. We present a machine learning approach to optimize aberration correction using a spatial light modulator. We implemented and compared Bayesian optimization and convolutional neural network (CNN) methods to predict optimal Zernike polynomial coefficients for wavefront correction. Our CNN achieved promising results with 80.39% accuracy on test data, demonstrating the potential for automated aberration correction in HHG systems.

physics.optics↗

arXiv2025

The CLT for lamplighter groups with an acylindrically hyperbolic base

We prove a Central Limit Theorem for the drift of a non-elementary random walk with a finite exponential moment on a wreath product $A\wr H=\bigoplus_{H} A\rtimes H$ with $A$ a non-trivial finite group and $H$ a finitely generated acylindrically hyperbolic group. We also provide the upper bounds on the central moments of the drift. Furthermore, our results extend to the case where $A$ is an arbitrary (possibly infinite) finitely generated group.

math.PR↗

arXiv2025

The Poisson boundary of Thompson's group $T$ is not the circle

Let $μ$ be a nondegenerate probability measure with finite entropy on a countable group $G \leq \mathrm{Homeo}_+(S^1)$ of orientation-preserving homeomorphisms of the circle acting proximally, minimally and topologically nonfreely on $S^1$. We prove that the circle $S^1$ endowed with its unique $μ$-stationary probability measure is not the Poisson boundary of $(G,μ)$. When $G$ is Thompson's group $T$ and $μ$ is finitely supported, this answers a question posed by B. Deroin [Ergodic Theory Dynam. Systems, 2013] and A. Navas [Proceedings of the International Congress of Mathematicians, 2018].

math.DS↗

arXiv2024

The Poisson boundary of lampshuffler groups

We study random walks on the lampshuffler group $\mathrm{FSym}(H)\rtimes H$, where $H$ is a finitely generated group and $\mathrm{FSym}(H)$ is the group of finitary permutations of $H$. We show that for any step distribution $μ$ with a finite first moment that induces a transient random walk on $H$, the permutation coordinate of the random walk almost surely stabilizes pointwise. Our main result states that for $H=\mathbb{Z}$, the above convergence completely describes the Poisson boundary of the random walk $(\mathrm{FSym}(\mathbb{Z})\rtimes \mathbb{Z},μ)$.

math.GR↗

arXiv2023

Dead ends on wreath products and lamplighter groups

For any finite group $A$ and any finitely generated group $B$, we prove that the corresponding lamplighter group $A\wr B$ admits a standard generating set with unbounded depth, and that if $B$ is abelian then the above is true for every standard generating set. This generalizes the case where $B=\mathbb{Z}$ together with its cyclic generator due to Cleary and Taback. When $B=H*K$ is the free product of two finite groups $H$ and $K$, we characterize which standard generators of the associated lamplighter group have unbounded depth in terms of a geometrical constant related to the Cayley graphs of $H$ and $K$. In particular, we find differences with the one-dimensional case: the lamplighter group over the free product of two sufficiently large finite cyclic groups has uniformly bounded depth with respect to some standard generating set.

math.GR↗

arXiv2021

Subshifts and colorings on ascending HNN-extensions of finitely generated abelian groups

For an ascending HNN-extension $G*_ψ$ of a finitely generated abelian group $G$, we study how a synchronization between the geometry of the group and weak periodicity of a configuration in $\mathcal{A}^{G*_ψ}$ forces global constraints on it, as well as in subshifts containing it. A particular case are Baumslag-Solitar groups $\mathrm{BS}(1,N)$, $N\ge2$, for which our results imply that a $\mathrm{BS}(1,N)$-SFT which contains a configuration with period $a^{N^\ell}$, $\ell\ge 0$, must contain a strongly periodic configuration with monochromatic $\mathbb{Z}$-sections. Then we study proper $n$-colorings, $n\ge 3$, of the (right) Cayley graph of $\mathrm{BS}(1,N)$, estimating the entropy of the associated subshift together with its mixing properties. We prove that $\mathrm{BS}(1,N)$ admits a frozen $n$-coloring if and only if $n=3$. We finally suggest generalizations of the latter results to $n$-colorings of ascending HNN-extensions of finitely generated abelian groups.

math.DS↗

arXiv2020

Random iterations of maps on $\mathbb{R}^{k}$: asymptotic stability, synchronization and functional central limit theorem

We study independent and identically distributed random iterations of continuous maps defined on a connected closed subset $S$ of the Euclidean space $\mathbb{R}^{k}$. We assume the maps are monotone (with respect to a suitable partial order) and a "topological" condition on the maps. Then, we prove the existence of a pullback random attractor whose distribution is the unique stationary measure of the random iteration, and we obtain the synchronization of random orbits. As a consequence of the synchronization phenomenon, a functional central limit theorem is established.

math.DS↗