Spherical metrics with conical singularities on 2-spheres

Suppose that $\theta_1,\theta_2,\dots,\theta_n$ are positive numbers and $n\ge 3$. Does there exist a sphere with a spherical metric with $n$ conical singularities of angles $2\pi\theta_1,2\pi\theta_2,\dots,2\pi\theta_n$? A sufficient condition was obtained by Gabriele Mondello and Dmitri Panov (arXiv:1505.01994). We show that it is also necessary when we assume that $\theta_1,\theta_2,\dots,\theta_n \not\in \mathbb{N}$.Comment: In this updated version, the proof of the main result has been changed, and is now more geometric in nature. 9 pages, 1 figur

A note on complex-hyperbolic Kleinian groups

Let $\Gamma$ be a discrete group of isometries acting on the complex hyperbolic $n$-space $\mathbb{H}^n_\mathbb{C}$. In this note, we prove that if $\Gamma$ is convex-cocompact, torsion-free, and the critical exponent $\delta(\Gamma)$ is strictly lesser than $2$, then the complex manifold $\mathbb{H}^n_\mathbb{C}/\Gamma$ is Stein. We also discuss several related conjectures.Comment: Minor revision. To appear in Arnold Math.

Patterson-Sullivan theory for Anosov subgroups

We extend several notions and results from the classical Patterson-Sullivan theory to the setting of Anosov subgroups of higher rank semisimple Lie groups, working primarily with invariant Finsler metrics on associated symmetric spaces. In particular, we prove the equality between the Hausdorff dimensions of flag limit sets, computed with respect to a suitable Gromov (pre-)metric on the flag manifold, and the Finsler critical exponents of Anosov subgroups.Comment: 46 pages. A gap in the proof of Theorem 8.3 in the earlier version has been fixed. Few other minor corrections mad

Borel Anosov subgroups of SL(d,R){\rm SL}(d,\mathbb{R})

We study the antipodal subsets of the full flag manifolds $\mathcal{F}(\mathbb{R}^d)$. As a consequence, for natural numbers $d \ge 2$ such that $d\ne 5$ and $d \not\equiv 0,\pm1 \mod 8$, we show that Borel Anosov subgroups of ${\rm SL}(d,\mathbb{R})$ are virtually isomorphic to either a free group or the fundamental group of a closed hyperbolic surface. Moreover, we obtain restrictions on the hyperbolic spaces admitting uniformly-regular quasiisometric embeddings into the symmetric space $X_d$ of ${\rm SL}(d,\mathbb{R})$.Comment: 20 pages, 1 figur

Klein-Maskit combination theorem for Anosov subgroups: Free products

We prove a generalization of the classical Klein-Maskit combination theorem, in the free product case, in the setting of Anosov subgroups. Namely, if $\Gamma_A$ and $\Gamma_B$ are Anosov subgroups of a semisimple Lie group $G$ of noncompact type, then under suitable topological assumptions, the group generated by $\Gamma_A$ and $\Gamma_B$ in $G$ is again Anosov, and is naturally isomorphic to the free product $\Gamma_A*\Gamma_B$. Such a generalization was conjectured in our previous article with Bernhard Leeb (arXiv:1805.07374).Comment: Final version after referee's comments, 26 pages. Accepted in Math.

Ping-pong in Hadamard manifolds

In this paper, we prove a quantitative version of the Tits alternative for negatively pinched manifolds $X$. Precisely, we prove that a nonelementary discrete isometry subgroup of $\mathrm{Isom}(X)$ generated by two non-elliptic isometries $g$, $f$ contains a free subgroup of rank $2$ generated by isometries $f^N , h$ of uniformly bounded word length. Furthermore, we show that this free subgroup is convex-cocompact when $f$ is hyperbolic.Comment: 20 pages, 5 figure

On the extended version of Krasnosel'skii's fixed point theorem for Kannan type equicontraction mappings

A sufficient condition is established for the existence of a solution to the equation $\mathcal{T}(u,\mathcal{C}(u))=u$, by considering a class of Kannan type equicontraction mappings $\mathcal{T}:\mathcal{A}\times \overline{\mathcal{C}(\mathcal{A})}\to \Xi$, where $\mathcal{A}$ is a convex, closed and bounded subset of a Banach space $\Xi$ and $\mathcal{C}$ is a compact mapping. To fulfil the desired purpose, we engage the Sadovskii's theorem, involving the measure of noncompactness. The relevance of the acquired results has been illustrated by considering a certain class of initial value problems

Restrictions on Anosov subgroups of Sp(2n,R)

Let $n\in\mathbb{N}$ and let $\Theta \subset \{1,\dots,n\}$ be a non-empty subset. We prove that if $\Theta$ contains an odd integer, then any $P_\Theta$-Anosov subgroup of ${\rm Sp}(2n,\mathbb{R})$ is virtually isomorphic to a free group or a surface group. In particular, any Borel Anosov subgroup of ${\rm Sp}(2n,\mathbb{R})$ is virtually isomorphic to a free or surface group. On the other hand, if $\Theta$ does not contain any odd integers, then there exists a $P_\Theta$-Anosov subgroup of ${\rm Sp}(2n,\mathbb{R})$ which is not virtually isomorphic to a free or surface group. We also exhibit new examples of maximally antipodal subsets of certain flag manifolds; these arise as limit sets of rank $1$ subgroups.Comment: 18 pages, 1 figur