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Register a new userOn classical uniformization theorems for higher dimensional complex Kleinian groups
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In this article we show that Bers simultaneous uniformization as well as the Koebes retrosection theorem are not longer true for discrete groups of projective transformations acting on the complex projective space.
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We study the periodic orbits problem on energy levels of Tonelli Lagrangian systems over configuration spaces of arbitrary dimension. We show that, when the fundamental group is finite and the Lagrangian has no stationary orbit at the Ma~ne critical energy level, there is a waist on every energy level just above the Ma~ne critical value. With a suitable perturbation with a potential, we show that there are infinitely many periodic orbits on every energy level just above the Ma~ne critical value, and on almost every energy level just below. Finally, we prove the Tonelli analogue of a closed geodesics result due to Ballmann-Thorbergsson-Ziller.
In this paper we generalize Kingmans sub-additive ergodic theorem to a large class of infinite countable discrete amenable group actions.
We compare different notions of limit sets for the action of Kleinian groups on the $n-$dimensional projective space via the irreducible representation $varrho:PSL(2,mathbb{C})to PSL(n+1,mathbb{C}).$ In particular, we prove that if the Kleinian group is convex-cocompact, the Myrberg and the Kulkarni limit coincide.
Let $G$ be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface $X subseteq mathbb{S}^2$. We prove that $G$ admits such an action that is in addition co-compact, provided we can replace $X$ by another surface $Y subseteq mathbb{S}^2$. We also prove that if a group $H$ has a finitely generated Cayley (multi-)graph $C$ covariantly embeddable in $mathbb{S}^2$, then $C$ can be chosen so as to have no infinite path on the boundary of a face. The proofs of these facts are intertwined, and the classes of groups they define coincide. In the orientation-preserving case they are exactly the (isomorphism types of) finitely generated Kleinian function groups. We construct a finitely generated planar Cayley graph whose group is not in this class. In passing, we observe that the Freudenthal compactification of every planar surface is homeomorphic to the sphere.
Given a discrete subgroup $Gamma$ of $PU(1,n)$ it acts by isometries on the unit complex ball $Bbb{H}^n_{Bbb{C}}$, in this setting a lot of work has been done in order to understand the action of the group. However when we look at the action of $Gamma$ on all of $ Bbb{P}^n_{Bbb{C}}$ little or nothing is known, in this paper study the action in the whole projective space and we are able to show that its equicontinuity agree with its Kulkarni discontuity set. Morever, in the non-elementary case, this set turns out to be the largest open set on which the group acts properly and discontinuously and can be described as the complement of the union of all complex projective hyperplanes in $ Bbb{P}^n_{Bbb{C}}$ which are tangent to $partial Bbb{H}^n_{Bbb{C}}$ at points in the Chen-Greenberg limit set $Lambda_{CG}(Gamma)$.
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