Geometric Structures and Representations of Discrete Groups

Kassel, Fanny

doi:10.1142/9789813272880_0090

Cited by 14 publications

(15 citation statements)

References 76 publications

(162 reference statements)

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“…Let be the conjugacy class of a parabolic subgroup . Then there is a natural notion of -Anosov representation ; see, for example, Kassel’s notes [34, Section 4]. When there is essentially one class , so we can simply refer to them as Anosov representations, and they can be defined as those injective representations where preserves and acts cocompactly on some nonempty convex subset V of X .…”

Section: Examplesmentioning

confidence: 99%

From curves to currents

Martínez-Granado

Thurston

2021

Forum of Mathematics, Sigma

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Many natural real-valued functions of closed curves are known to extend continuously to the larger space of geodesic currents. For instance, the extension of length with respect to a fixed hyperbolic metric was a motivating example for the development of geodesic currents. We give a simple criterion on a curve function that guarantees a continuous extension to geodesic currents. The main condition of our criterion is the smoothing property, which has played a role in the study of systoles of translation lengths for Anosov representations. It is easy to see that our criterion is satisfied for almost all known examples of continuous functions on geodesic currents, such as nonpositively curved lengths or stable lengths for surface groups, while also applying to new examples like extremal length. We use this extension to obtain a new curve counting result for extremal length.

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Section: Examplesmentioning

confidence: 99%

From curves to currents

Martínez-Granado

Thurston

2021

Forum of Mathematics, Sigma

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“…It is expected that not all linear hyperbolic group admit linear Anosov representations, but we know of no explicit examples. See also the discussion in Kassel [37, Section 8]. Question Can one exhibit explicit examples of linear hyperbolic groups which do not admit Anosov representations into

SL (d, R)

for any

d

?…”

Section: Examples and Questionsmentioning

confidence: 99%

“…We will use a recent theorem of Kassel–Potrie [38] (see also [37, Theorem 4.3]) to give a simple definition of

P_{k}

‐Anosov representations. If

normalΓ

is a hyperbolic group and

1 ⩽ k ⩽ \frac{d}{2}

, a representation

ρ : Γ \to GL (d, R)

is said to be

P_{k}

‐Anosov if there exist constants

μ, C > 0

so that

\frac{λ_{k} false(ρ (γ) false)}{λ_{k + 1} false(ρ (γ) false)} ⩾ C e^{μ | | γ | |}

for all

γ \in Γ

.…”

Section: Introductionmentioning

confidence: 99%

Topological restrictions on Anosov representations

Canary

Konstantinos

2020

Journal of Topology

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“…As beautifully described in Wienhard's 2018 ICM Proceedings article [Wie18], higher Teichmüller theory recently emerged as a new field in mathematics, closely related to Higgs bundles (see also [Kas18,Col19] Recall that a representation ρ : π 1 (Σ) → G is maximal if it maximizes the Toledo invariant T (ρ), a topological invariant defined for any simple Lie group G of Hermitian type as…”

Section: Higher Teichmüller Theorymentioning

confidence: 99%

“…see [Col19]). For an excellent review of the theory of geometric structures, the reader should refer to Kassel's 2018 ICM Proceedings article [Kas18].…”

Section: Higher Teichmüller Theorymentioning

confidence: 99%