We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called weakly maximal representations. We prove that weakly maximal representations are discrete and injective and we describe the structure of the Zariski closure of their image. Furthermore, we prove that the set of weakly maximal representations is a closed subset of the representation variety and describe its relation to other geometrically significant subsets of the representations variety.
M. B. was partially supported by the Swiss National Science Foundation project 200020-144373; T. H. was partially supported by the Swiss National Science Foundation project 2000021-127016/2; A. I. was partially supported by the Swiss National Science Foundation projects 2000021-127016/2 and 200020-144373; A. W. was partially supported by the National Science Foundation under agreement No. DMS-1065919 and 0846408, by the Sloan Foundation, by the Deutsche Forschungsgemeinschaft, and by the ERCEA under ERC-Consolidator grant no. 614733. Support by the Institut Mittag-Leffler (Djursholm, Sweden) and by the Institute for Advanced Study (Princeton, NJ) is gratefully acknowledged.
Gabi Ben Simon. Marc Burger. Tobias Hartnick. Alessandra Iozzi. Anna Wienhard. "On weakly maximal representations of surface groups." J. Differential Geom. 105 (3) 375 - 404, March 2017. https://doi.org/10.4310/jdg/1488503002