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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.
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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.
We consider exact sequences and lower central series of surface braid groups and we explain how they can prove to be useful for obtaining representations for surface braid groups. In particular, using a completely algebraic framework, we describe the notion of extension of a representation introduced and studied recently by An and Ko and independently by Blanchet.
In this paper, we study the geometric and dynamical properties of maximal representations of surface groups into Hermitian Lie groups of rank 2. Combining tools from Higgs bundle theory, the theory of Anosov representations, and pseudo-Riemannian geometry, we obtain various results of interest. We prove that these representations are holonomies of certain geometric structures, recovering results of Guichard and Wienhard. We also prove that their length spectrum is uniformly bigger than that of a suitably chosen Fuchsian representation, extending a previous work of the second author. Finally, we show that these representations preserve a unique minimal surface in the symmetric space, extending a theorem of Labourie for Hitchin representations in rank 2.
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.
A beautifully simple free generating set for the commutator subgroup of a free group was constructed by Tomaszewski. We give a new geometric proof of his theorem, and show how to give a similar free generating set for the commutator subgroup of a surface group. We also give a simple representation-theoretic description of the structure of the abelianizations of these commutator subgroups and calculate their homology.
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