arXiv subjects

Eduardo Oregón-Reyes

Publications and source records attributed to Eduardo Oregón-Reyes.

5 recordsLinked to original sources

arXiv2022

The space of metric structures on hyperbolic groups

We study the metric and topological properties of the space $\mathscr{D}(G)$ of left-invariant hyperbolic pseudometrics on the non-elementary hyperbolic group $G$ that are quasi-isometric to a word metric, up to rough similarity. This space naturally contains the Teichmüller space in case $G$ is a surface group and the Culler-Vogtmann outer space when $G$ is a free group. Endowed with a natural metric reminiscent of the (symmetrized) Thurston's metric on Teichmüller space, we prove that $\mathscr{D}(G)$ is an unbounded contractible metric space and that $\mathrm{Out}(G)$ acts metrically properly by isometries on it. If we restrict ourselves to the subspace $\mathscr{D}_δ(G)$ of the points represented by $δ$-hyperbolic metrics with critical exponent 1, we prove that it is either empty or proper. We also prove continuity of the Bowen-Margulis map from $\mathscr{D}_δ(G)$ into the space $\mathbb{P}\mathcal{C}urr(G)$ of projective geodesic currents on $G$, extending similar results for surface and free groups, and the continuity of the (normalized) mean distortion as a function on $\mathscr{D}(G)\times \mathscr{D}(G)$.

math.GR↗

arXiv2022

On cubulated relatively hyperbolic groups

We show that properly and cocompactly cubulated relatively hyperbolic groups are virtually special, provided the peripheral subgroups are virtually special in a way that is compatible with the cubulation. This extends Agol's result for cubulated hyperbolic groups, and applies to a wide range of peripheral subgroups. In particular, we deduce virtual specialness for properly and cocompactly cubulated groups that are hyperbolic relative to virtually abelian groups. As another consequence, by using a theorem of Martin and Steenbock we obtain virtual specialness for groups obtained as a quotient of a free product of finitely many virtually compact special groups by a finite set of relators satisfying the classical $C'(1/6)$-small cancellation condition.

math.GR↗

arXiv2019

A new inequality about matrix products and a Berger-Wang formula

We prove an inequality relating the norm of a product of matrices $A_n\cdots A_1$ with the spectral radii of subproducts $A_j\cdots A_i$ with $1\leq i\leq j\leq n$. Among the consequences of this inequality, we obtain the classical Berger-Wang formula as an immediate corollary, and give an easier proof of a characterization of the upper Lyapunov exponent due to I. Morris. As main ingredient for the proof of this result, we prove that for a large enough $n$, the product $A_n\cdots A_1$ is zero under the hypothesis that $A_j\cdots A_i$ are nilpotent for all $1\leq i \leq j\leq n$.

math.DS↗

arXiv2019

The Avalanche Principle and negative curvature

We use the geometric structure of the hyperbolic upper half plane to provide a new proof of the Avalanche Principle introduced by M. Goldstein and W. Schlag in the context of $\mathrm{SL}_{2}(\mathbb{R})$ matrices. This approach allows to interpret and extend this result to arbitrary $\mathrm{CAT}(-1)$ metric spaces. Through the proof, we deduce a polygonal Schur theorem for these spaces.

math.MG↗

arXiv2018

Properties of sets of isometries of Gromov hyperbolic spaces

We prove an inequality concerning isometries of a Gromov hyperbolic metric space, which does not require the space to be proper or geodesic. It involves the joint stable length, a hyperbolic version of the joint spectral radius, and shows that sets of isometries behave like sets of $2 \times 2$ real matrices. Among the consequences of the inequality, we obtain the continuity of the joint stable length and an analogue of Berger-Wang theorem.

math.MG↗