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Register a new userThe limit set of discrete subgroups of $PSL(3,C)$
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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.
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In this article we provide an algebraic characterization of those groups of $PSL(3,Bbb{C})$ whose limit set in the Kulkarni sense has, exactly, four lines in general position. Also we show that, for this class of groups, the equicontinuity set of the group is the largest open set where the group acts discontinuously and agrees with the discontinuity set of the group.
In this note, we show that the exceptional algebraic set of an infinite discrete group in $PSL(3,Bbb{C})$ should be a finite union of complex lines, copies of the Veronese curve or copies of the cubic $xy^2-z^3$.
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.
Let $ G $ be a discrete subgroup of PU(1,n). Then $ G $ acts on $mathbb {P}^n_mathbb C$ preserving the unit ball $mathbb {H}^n_mathbb {C}$, where it acts by isometries with respect to the Bergman metric. In this work we determine the equicontinuty region $Eq(G)$ of $G$ in $mathbb P^n_{mathbb C}$: It is the complement of the union of all complex projective hyperplanes in $mathbb {P}^n_{mathbb C}$ which are tangent to $partial mathbb {H}^n_mathbb {C}$ at points in the Chen-Greenberg limit set $Lambda_{CG}(G )$, a closed $G$-invariant subset of $partial mathbb {H}^n_mathbb {C}$, which is minimal for non-elementary groups. We also prove that the action on $Eq(G)$ is discontinuous.
We study the local equivalence problem for real-analytic ($mathcal{C}^omega$) hypersurfaces $M^5 subset mathbb{C}^3$ which, in coordinates $(z_1, z_2, w) in mathbb{C}^3$ with $w = u+i, v$, are rigid: [ u ,=, Fbig(z_1,z_2,overline{z}_1,overline{z}_2big), ] with $F$ independent of $v$. Specifically, we study the group ${sf Hol}_{sf rigid}(M)$ of rigid local biholomorphic transformations of the form: [ big(z_1,z_2,wbig) longmapsto Big( f_1(z_1,z_2), f_2(z_1,z_2), a,w + g(z_1,z_2) Big), ] where $a in mathbb{R} backslash {0}$ and $frac{D(f_1,f_2)}{D(z_1,z_2)} eq 0$, which preserve rigidity of hypersurfaces. After performing a Cartan-type reduction to an appropriate ${e}$-structure, we find exactly two primary invariants $I_0$ and $V_0$, which we express explicitly in terms of the $5$-jet of the graphing function $F$ of $M$. The identical vanishing $0 equiv I_0 big( J^5F big) equiv V_0 big( J^5F big)$ then provides a necessary and sufficient condition for $M$ to be locally rigidly-biholomorphic to the known model hypersurface: [ M_{sf LC} colon u ,=, frac{z_1,overline{z}_1 +frac{1}{2},z_1^2overline{z}_2 +frac{1}{2},overline{z}_1^2z_2}{ 1-z_2overline{z}_2}. ] We establish that $dim, {sf Hol}_{sf rigid} (M) leq 7 = dim, {sf Hol}_{sf rigid} big( M_{sf LC} big)$ always. If one of these two primary invariants $I_0 otequiv 0$ or $V_0 otequiv 0$ does not vanish identically, we show that this rigid equivalence problem between rigid hypersurfaces reduces to an equivalence problem for a certain $5$-dimensional ${e}$-structure on $M$.
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