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Register a new userExceptional Algebraic sets for discrete groups of $PSL(3,Bbb{C})$
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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 $xy^2-z^3$.
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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.
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.
Let $G$ be a simple algebraic group over an algebraically closed field $k$ and let $C_1, ldots, C_t$ be non-central conjugacy classes in $G$. In this paper, we consider the problem of determining whether there exist $g_i in C_i$ such that $langle g_1, ldots, g_t rangle$ is Zariski dense in $G$. First we establish a general result, which shows that if $Omega$ is an irreducible subvariety of $G^t$, then the set of tuples in $Omega$ generating a dense subgroup of $G$ is either empty or dense in $Omega$. In the special case $Omega = C_1 times cdots times C_t$, by considering the dimensions of fixed point spaces, we prove that this set is dense when $G$ is an exceptional algebraic group and $t geqslant 5$, assuming $k$ is not algebraic over a finite field. In fact, for $G=G_2$ we only need $t geqslant 4$ and both of these bounds are best possible. As an application, we show that many faithful representations of exceptional algebraic groups are generically free. We also establish new results on the topological generation of exceptional groups in the special case $t=2$, which have applications to random generation of finite exceptional groups of Lie type. In particular, we prove a conjecture of Liebeck and Shalev on the random $(r,s)$-generation of exceptional groups.
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.
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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