Maximal representations in lattices of the symplectic group
Jacques Audibert
TL;DR
This paper demonstrates that most lattices in the symplectic group contain infinitely many Zariski-dense maximal representations, revealing new structures and deformations within these lattices, especially highlighting the presence of surface subgroups.
Contribution
It establishes the existence of infinitely many Zariski-dense maximal representations in lattices of Sp(2n,R), except for specific cases, and shows that these include surface subgroups.
Findings
Most lattices in Sp(2n,R) contain infinitely many Zariski-dense maximal representations.
Sp(4k,Z) contains Zariski-dense surface subgroups for all k.
Except for lattices commensurable with Sp(4k+2,Z) when n=2k+1, all other lattices contain these representations.
Abstract
We prove that all lattices of Sp(2n,R), except those commensurable with Sp(4k+2,Z) when n=2k+1, contain the image of infinitely many mapping class group orbits of Zariski-dense maximal representation that are continuous deformations of maximal diagonal representations. In particular, we show that Sp(4k,Z) contain Zariski-dense surface subgroups for all k.
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Taxonomy
TopicsAdvanced Algebra and Geometry · Geometric and Algebraic Topology · Finite Group Theory Research