arXiv subjects

Alonso Beaumont

Publications and source records attributed to Alonso Beaumont.

3 recordsLinked to original sources

arXiv2025

On the centralizers of endomorphisms of the projective line

Let $f$ be a dominant endomorphism of the projective line, which is not conjugate to a power map $z\mapsto z^{\pm d}$. We consider the centralizers of the iterates of $f$, $C(f^{n}):=\{\textrm{dominant}\;g:\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}\;|\; g\circ f^{n}=f^{n}\circ g\}$, $n\geq1$, and prove that their union is equal to $C(f^{N})$ for some $N\geq1$. This solves a conjecture of F. Pakovich. As an application, we obtain a Tits alternative for cancellative semigroups of endomorphisms of the projective line, without an assumption of finite generation, extending the results of J.P. Bell, K. Huang, W. Peng and T.J. Tucker.

math.DS↗

arXiv2025

A uniform Tits alternative for endomorphisms of the projective line

A recent article of J.P. Bell, K. Huang, W. Peng and T.J. Tucker establishes an analog of the Tits alternative for semigroups of endomorphisms of the projective line. The proof involves a ping-pong argument on arithmetic height functions. Extending this method, we obtain a uniform version of the same alternative. In particular, we show that semigroups of $\mathrm{End}(\mathbb{P}^{1})$ of exponential growth are of uniform exponential growth.

math.NT↗

arXiv2024

On the Nilpotency of Locally Pro-p Contraction Groups

H. Gl\"ockner and G. A. Willis have recently shown that locally pro-p contraction groups are nilpotent. The proof hinges on a fixed-point result: if the local field $\mathbb{F}_{p}(\!(t)\!)$ acts on its $d$-th power $\mathbb{F}_{p}(\!(t)\!)^{d}$ additively, continuously, and in an appropriately equivariant manner, then the action has a non-zero fixed point. We provide a short proof of this theorem.

math.GR↗