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In this paper we study Zimmer's conjecture for $$C^{1}$$ actions of lattice subgroup of a higher-rank simple Lie group with finite center on compact manifolds. We show that when the rank of an uniform lattice is larger than the dimension of the manifold, then the action factors through a finite group. For lattices in $${\rm SL}(n, {{\mathbb {R}}})$$ , the dimensional bound is sharp.
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Real Slices of ${\rm SL}(r,\mathbb{C})$-Opers
Through the action of an anti-holomorphic involution $$\sigma$$ (a real structure) on a Riemann surface $$X$$, we consider the induced actions on $${\rm SL}(r,\mathbb{C})$$-opers and study the real slices fixed by such actions. By constructing this involution for different descriptions of the space of $${\rm SL}(r,\mathbb{C})$$-opers, we are able to give a natural parametrization of the fixed point locus via differentials on the Riemann surface, which in turn allows us to study their geometric properties.
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- PAR ID:
- 10500392
- Publisher / Repository:
- SIGMA
- Date Published:
- Journal Name:
- Symmetry, Integrability and Geometry: Methods and Applications
- ISSN:
- 1815-0659
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3842/SIGMA.2023.067
