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We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called weakly maximal representations. They are defined in terms of invariants in bounded cohomology and extend considerably the scope of maximal representations. We prove that weakly maximal representations are discrete and injective and describe the structure of the Zariski closure of the image. An interesting feature of these representations is that they admit an elementary topological characterization in terms of bi-invariant orderings. In particular if the target group is Hermitian of tube type, the ordering can be described in terms of the causal structure on the Shilov boundary.
These are the lecture notes from my course in the January 2011 School on Moduli Spaces at the Newton Institute. I give an introduction to Higgs bundles and their application to the study of character varieties for surface group representations.
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 geo
We prove a generalization of a conjecture of C. Marion on generation properties of finite groups of Lie type, by considering geometric properties of an appropriate representation variety and associated tangent spaces.
We show that for a fixed k, Gromov random groups with any positive density have no non-trivial degree-k representations over any field, a.a.s. This is especially interesting in light of the results of Agol, Ollivier and Wise that when the density is
In 1933 B.~H.~Neumann constructed uncountably many subgroups of ${rm SL}_2(mathbb Z)$ which act regularly on the primitive elements of $mathbb Z^2$. As pointed out by Magnus, their images in the modular group ${rm PSL}_2(mathbb Z)cong C_3*C_2$ are ma