arXiv subjects
Publications and source records attributed to Emiliano Sequeira.
We study the volume growth of horospheres in a Heintze group of the form R ___ A R d with A a diagonal derivation. We conclude that the isometry and quasi-isometry classes of horospheres (with their intrinsic geometry) coincide. Furthermore, if A is not a scalar multiple of the identity, then there are exactly two such classes, characterized by their volume growth, which we calculate explicitly.
We generalize some results on asymptotic and continuous group $L^p$-cohomology to Orlicz cohomology. In particular, we show that asymptotic Orlicz cohomology is a quasi-isometry invariant and that both notions coincide in the case of a locally compact second countable group. The case of degree $1$ is studied in more detail.
We introduce the notion of \textit{relative $L^p$-cohomology} as a quasi-isometry invariant defined for Gromov-hyperbolic spaces, and apply it to the problem of quasi-isometry classification of Heintze groups. More precisely, we explicitly construct non-zero relative $L^p$-cohomology classes on a Heintze group of the form $\mathbb{R}^{n-1}\rtimes_α\mathbb{R}$, which gives a way to prove that the eigenvalues of $α$, up to a scalar multiple, are invariant by quasi-isometries. In the case of degree $1$ we show a relation between the relative and the classical $L^p$-cohomology.
We prove that the de Rham $L^ϕ$-cohomology of a Riemannian manifold $M$ admiting a convenient triangulation $X$ is isomorphic to the simplicial $\ell^ϕ$-cohomology of $X$ for any Young function $ϕ$. This result implies the quasi-isometry invariance of the first one.
In this work we reproduce the characterization of $\Gg^s$-sets from the euclidean setting [J. London Math. Soc. 49:267-280,1994] to more general metric spaces. These sets have Hausdorff dimension at least $s$ and are closed by countable intersections, which is particularly useful to estimate the dimension of the so called sets of $α$-approximable points (that typically appear in Diophantine approximations).
We prove the equivalence between the simplicial Orlicz cohomology and the Orlicz-de Rham cohomology in the case of Lie groups. Since the first one is a quasi-isometry invariant for uniformly contractible simplicial complexes with bounded geometry, we obtain the invariance of the second one in the case of contractible Lie groups. We also define the Orlicz cohomology of a Gromov-hyperbolic space relative to a point on its boundary at infinity, for which the same results are true.
A a Heintze group is a Lie group of the form $N\rtimes_α\mathbb{R}$, where $N$ is a simply connected nilpotent Lie group and $α$ is a derivation of $\mathrm{Lie}(N)$ whose eigenvalues all have positive real parts. We show that if two purely real Heintze groups equipped with left-invariant metrics are quasi-isometric, then up to a positive scalar multiple, their respective derivations have the same characteristic polynomial. Using the same thecniques, we prove that if we restrict to the class of Heintze groups for which $N$ is the Heisenberg group, then the Jordan form of $α$, up to positive scalar multiples, is a quasi-isometry invariant.