Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDiscrete subgroups of small critical exponent
154
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We prove that finitely generated higher dimensional Kleinian groups with small critical exponent are always convex-cocompact. Along the way, we also prove some geometric properties for any complete pinched negatively curved manifold with critical exponent less than 1.
rate research
Read More
In arXiv:1802.02833 Guichard and Wienhard introduced the notion of $Theta$-positivity, a generalization of Lusztigs total positivity to real Lie groups that are not necessarily split. Based on this notion, we introduce in this paper $Theta$-positive representations of surface groups. We prove that $Theta$-positive representations are $Theta$-Anosov. This implies that $Theta$-positive representations are discrete and faithful and that the set of $Theta$-positive representations is open in the representation variety. We show that the set of $Theta$-positive representations is closed within the set of representations that do not virtually factor through a parabolic subgroup. From this we deduce that for any simple Lie group $mathsf G$ admitting a $Theta$-positive structure there exist components consisting of $Theta$-positive representations. More precisely we prove that the components parametrized using Higgs bundles methods in arXiv:2101.09377 consist of $Theta$-positive representations.
We prove a theorem that gives a sufficient condition for the full basic automorphism group of a complete Cartan foliation to admit a unique (finite-dimensional) Lie group structure in the category of Cartan foliations. Emphasize that the transverse Cartan geometry may not be effective. Some estimates of the dimension of this group depending on the transverse geometry are found. Further, we investigate Cartan foliations covered by fibrations. When the global holonomy group of that foliation is discrete, we obtain the explicit new formula for determining its basic automorphism Lie group. Examples of computing the full basic automorphism group of complete Cartan foliations are constructed.
Let $X$ be a proper geodesic Gromov hyperbolic metric space and let $G$ be a cocompact group of isometries of $X$ admitting a uniform lattice. Let $d$ be the Hausdorff dimension of the Gromov boundary $partial X$. We define the critical exponent $delta(mu)$ of any discrete invariant random subgroup $mu$ of the locally compact group $G$ and show that $delta(mu) > frac{d}{2}$ in general and that $delta(mu) = d$ if $mu$ is of divergence type. Whenever $G$ is a rank-one simple Lie group with Kazhdans property $(T)$ it follows that an ergodic invariant random subgroup of divergence type is a lattice. One of our main tools is a maximal ergodic theorem for actions of hyperbolic groups due to Bowen and Nevo.
If $Gamma$ is a discrete subgroup of $PSL(3,Bbb{C})$, it is determined the equicontinuity region $Eq(Gamma)$ of the natural action of $Gamma$ on $Bbb{P}^2_Bbb{C}$. It is also proved that the action restricted to $Eq(Gamma)$ is discontinuous, and $Eq(Gamma)$ agrees with the discontinuity set in the sense of Kulkarni whenever the limit set of $Gamma$ in the sense of Kulkarni, $Lambda(Gamma)$, contains at least three lines in general position. Under some additional hypothesis, it turns out to be the largest open set on which $Gamma$ acts discontinuously. Moreover, if $Lambda(Gamma)$ contains at least four complex lines and $Gamma$ acts on $Bbb{P}^2_Bbb{C}$ without fixed points nor invariant lines, then each connected component of $Eq(Gamma)$ is a holomorphy domain and a complete Kobayashi hyperbolic space.
We introduce $Theta$-positivity, a new notion of positivity in real semisimple Lie groups. The notion of $Theta$-positivity generalizes at the same time Lusztigs total positivity in split real Lie groups as well as well known concepts of positivity in Lie groups of Hermitian type. We show that there are two other families of Lie groups, SO(p,q) for p<q, and a family of exceptional Lie groups, which admit a $Theta$-positive structure. We describe key aspects of $Theta$-positivity and make a connection with representations of surface groups and higher Teichmuller theory.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
