arXiv subjects
Publications and source records attributed to Maximiliano Escayola.
Let $n\ge 2$, and let $M$ be either the $n$-dimensional sphere $\mathbb{S}^n$ or the real projective space $\mathbb{RP}^n$. Given $α\ge 1$, we construct a finitely generated group $G_{n,α}$ that embeds into the group of $C^α$-diffeomorphisms of $M$, but does not embed into the group of $C^β$-diffeomorphisms of $M$ for any $β>α$.
We investigate acylindrically hyperbolic groups acting on the circle by piecewise linear homeomorphisms. We prove that a finitely generated acylindrically hyperbolic group acting faithfully by piecewise linear homeomorphisms of the circle is virtually free, is virtually a closed hyperbolic surface group, or virtually splits over a two-ended subgroup; as a consequence, we prove a general structural result for Gromov hyperbolic groups of piecewise linear homeomorphisms. Along the way, we show that a right-angled Artin subgroup of piecewise linear homeomorphisms of the circle is either free or abelian, generalizing a result of Bleak and Salazar-Diaz. We also show that the fundamental group of a finite volume hyperbolic $n$--manifold cannot act faithfully on the circle by piecewise linear homeomorphisms, provided that $n\geq 3$, complementing $2$--dimensional examples of Ghys and Minakawa.
We construct a $C^1$-diffeomorphism of the circle having an invariant Cantor minimal set such that, when the lengths of the connected components of its complement are arranged in decreasing order, the ratios of consecutive lengths do not converge to 1. This gives a negative answer to a longstanding question posed by Dusa McDuff.
Given a finitely generated, torsion-free nilpotent group, we find the maximum possible (critical) regularity for its faithful actions by diffeomorphisms of the closed or half-open interval and of the circle. Our result gives an expression for its value in purely algebraic terms (using the relative growth of appropriate subgroups), generalizing many preceding works. As an intermediate step, we generalize the Bass-Guivarc'h formula, obtaining a formula for the relative growth of subgroups of nilpotent groups, as well as for the growth of the corresponding Schreier graphs.
Let $G$ be a torsion-free, finitely-generated, nilpotent and metabelian group. In this work we show that $G$ embeds into the group of orientation preserving $C^{1+α}$-diffeomorphisms of the compact interval, for all $α< 1/k$ where $k$ is the torsion-free rank of $G/A$ and $A$ is a maximal abelian subgroup. We show that in many situations the corresponding $1/k$ is critical in the sense that there is no embedding of $G$ with higher regularity. A particularly nice family where this happens, is the family of $(2n+1)$-dimensional Heisenberg groups, for which we can show that the critical regularity equals $1+1/n$.
We improve a recent construction of Andrés Navas to produce the first examples of $C^2$-undistorted diffeomorphisms of the interval that are $C^{1+α}$-distorted (for every $α< 1$). We do this via explicit computations due to the failure of an extension to class $C^{1+α}$ of a classical lemma related to the work of Nancy Kopell.