Schottky groups and maximal representations

Burelle, Jean-Philippe; Treib, Nicolaus

doi:10.1007/s10711-017-0285-2

Cited by 5 publications

(9 citation statements)

References 25 publications

(26 reference statements)

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“…In this section we recall definitions from [BT18] involving cyclic orders and the definition of a Schottky group in a cyclically ordered space.…”

Section: Partially Cyclically Ordered Spacesmentioning

confidence: 99%

“…In [BT18], the main theorem is the existence of an equivariant boundary map for a certain class of generalized Schottky groups :…”

Section: Partially Cyclically Ordered Spacesmentioning

confidence: 99%

“…We now turn to proving a list of useful properties of the partial cyclic order on Flag + (R n ) given by hyperconvexity. These properties will allow us to construct Schottky representations and use the corresponding results from [BT18].…”

Section: The Partial Cyclic Ordermentioning

confidence: 99%

“…Although the tools used to study maximal representations are generally different from those which were successful for Hitchin components and positive representations, the one feature which seems to connect these higher Teichmüller theories is the cyclic structure on the boundary of the group, and a compatible cyclic structure on a homogeneous space of G (see [BIW14] for details on this analogy). In [BT18], the authors defined a notion of generalized Schottky group of automorphisms of a space admitting a partial cyclic order, and showed that maximal representations are examples of this construction. This characterization was then used to build fundamental domains for the action of maximal representations into symplectic groups Sp(2n, R) on a domain of discontinuity in projective space.…”

Section: Introductionmentioning

confidence: 99%

“…In order emphasize the similarities between positive and maximal representations, we have structured the paper in a similar way to the previous paper [BT18].…”

Section: Introductionmentioning

confidence: 99%

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Schottky presentations of positive representations

Burelle¹,

Treib²

2018

Preprint

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We show that the notion of 3-hyperconvexity on oriented flag manifolds defines a partial cyclic order. Using the notion of interval given by this partial cyclic order, we construct Schottky groups and show that they correspond to images of positive representations in the sense of Fock and Goncharov. We construct polyhedral fundamental domains for the domain of discontinuity that these groups admit in the projective space or the sphere, depending on the dimension.

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“…In this section we recall definitions from [BT18] involving cyclic orders and the definition of a Schottky group in a cyclically ordered space.…”

Section: Partially Cyclically Ordered Spacesmentioning

confidence: 99%

“…In [BT18], the main theorem is the existence of an equivariant boundary map for a certain class of generalized Schottky groups :…”

Section: Partially Cyclically Ordered Spacesmentioning

confidence: 99%

Section: The Partial Cyclic Ordermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In order emphasize the similarities between positive and maximal representations, we have structured the paper in a similar way to the previous paper [BT18].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Schottky presentations of positive representations

Burelle¹,

Treib²

2018

Preprint

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Schottky presentations of positive representations

Burelle

Treib

2022

Math. Ann.

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Domains of discontinuity in oriented flag manifolds

Stecker

Treib

2022

Journal of London Math Soc

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We study actions of discrete subgroups Γ$\Gamma$ of semi‐simple Lie groups G$G$ on associated oriented flag manifolds. These are quotients G/P$G/P$, where the subgroup P$P$ lies between a parabolic subgroup and its identity component. For Anosov subgroups normalΓ⊂G$\Gamma \subset G$, we identify domains in oriented flag manifolds by removing a set obtained from the limit set of Γ$\Gamma$, and give a combinatorial description of proper discontinuity and cocompactness of these domains. This generalizes analogous results of Kapovich–Leeb–Porti to the oriented setting. We give first examples of cocompact domains of discontinuity which are not lifts of domains in unoriented flag manifolds. These include in particular domains in oriented Grassmannians for Hitchin representations, which we also show to be nonempty. As a further application of the oriented setup, we give a new lower bound on the number of connected components of B$B$‐Anosov representations of a closed surface group into prefixSLfalse(n,double-struckRfalse)$\operatorname{SL}(n,\mathbb {R})$.

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Schottky groups and maximal representations

Cited by 5 publications

References 25 publications

Schottky presentations of positive representations

Schottky presentations of positive representations

Schottky presentations of positive representations

Domains of discontinuity in oriented flag manifolds

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