Counting, equidistribution and entropy gaps at infinity with applications to cusped Hitchin representations

Bray, Harrison; Canary, Richard D.; Kao, Lien-Yung; Martone, Giuseppe

doi:10.48550/arxiv.2102.08552

Cited by 7 publications

(25 citation statements)

References 52 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this proof, we will use the technology developed in Bray-Canary-Kao-Martone [11]. We note that the results of that paper were stated for representations into SL(d, R), but a careful reading verifies that the same arguments taken verbatim work for representations into PGL(d, R).…”

Section: Quint's Indicator Setmentioning

confidence: 89%

“…Remark. To be precise, Corollary 11.1 in [11] was stated for representations into SL(d, R), but the same argument taken verbatim works for representations into PGL(d, K) since the construction of the roof functions only involve the Cartan projection.…”

Section: 2mentioning

confidence: 98%

“…Notice that part (3) is an immediate consequence of the definition and Lemma 2.1. Bray, Canary, Kao and Martone [11] established counting and equidistribution results for cusped Anosov representations. We will need the following counting result, which generalizes a result of Sambarino [45, Thm.…”

Section: 2mentioning

confidence: 98%

“…Notice that Theorem 2.2 implies that (a * θ ) + ⊂ B + θ (ρ) whenever ρ is P θ -Anosov. Theorem 2.3 (Bray-Canary-Kao-Martone [11,Corollary 11.1]). Suppose that Γ is a torsion-free, geometrically finite Fuchsian group and ρ :…”

Section: 2mentioning

confidence: 99%

“…where [Γ hyp ] is the set of conjugacy classes of hyperbolic elements of Γ. When ρ is a Hitchin representation of a geometrically finite Fuchsian group, the lim sup in the definition of h φ (ρ) holds as a limit, and is always positive and finite (see [11]).…”

mentioning

confidence: 99%

See 4 more Smart Citations

Entropy rigidity for cusped Hitchin representations

Canary¹,

Zhang²,

Zimmer³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

We establish an entropy rigidity theorem for Hitchin representations of all geometrically finite Fuchsian groups which generalizes a theorem of Potrie and Sambarino for Hitchin representations of closed surface groups. In the process, we introduce the class of (1, 1, 2)hypertransverse groups and show for such a group that the Hausdorff dimension of its conical limit set agrees with its (first) simple root entropy, providing a common generalization of results of Bishop and Jones, for Kleinian groups, and Pozzetti, Sambarino and Wienhard, for Anosov groups. We also introduce the theory of transverse representations of projectively visible groups as a tool for studying discrete subgroups of linear groups which are not necessarily Anosov or relatively Anosov.

show abstract

Section: Quint's Indicator Setmentioning

confidence: 89%

Section: 2mentioning

confidence: 98%

Section: 2mentioning

confidence: 98%

Section: 2mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Entropy rigidity for cusped Hitchin representations

Canary¹,

Zhang²,

Zimmer³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Cataclysms for Anosov representations

Pfeil

2022

Geom Dedicata

View full text Add to dashboard Cite

In this paper, we construct cataclysm deformations for $$\theta $$ θ -Anosov representations into a semisimple non-compact connected real Lie group G with finite center, where $$\theta \subset \Delta $$ θ ⊂ Δ is a subset of the simple roots that is invariant under the opposition involution. These generalize Thurston’s cataclysms on Teichmüller space and Dreyer’s cataclysms for Borel-Anosov representations into $$\mathrm {PSL}(n, \mathbb {R})$$ PSL ( n , R ) . We express the deformation also in terms of the boundary map. Furthermore, we show that cataclysm deformations are additive and behave well with respect to composing a representation with a group homomorphism. Finally, we show that the deformation is injective for Hitchin representations, but not in general for $$\theta $$ θ -Anosov representations.

show abstract

Cusped Hitchin representations and Anosov representations of geometrically finite Fuchsian groups

Canary¹,

Zhang²,

Zimmer³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We develop a theory of Anosov representation of geometrically finite Fuchsian groups in SL(d, R) and show that cusped Hitchin representations are Borel Anosov in this sense. We establish analogues of many properties of traditional Anosov representations. In particular, we show that our Anosov representations are stable under type-preserving deformations and that their limit maps vary analytically. We also observe that our Anosov representations fit into the previous frameworks of relatively Anosov and relatively dominated representations developed by Kapovich-Leeb and Zhu. Contents 1. Introduction 1 2. Preliminaries 4 3. Anosov representations into SL(d, K) 6 4. Basic properties of Anosov representations 9 5. Basic properties of cusp representations 13 6. A dynamical characterization of linear Anosov representations 15 7. Hitchin representations are Borel Anosov 20 8. Stability of Anosov representations 22 9. Positive representations in the sense of Fock-Goncharov 29 Appendix A. Anosov representations into semisimple Lie groups 33 References 38

show abstract

Counting, equidistribution and entropy gaps at infinity with applications to cusped Hitchin representations

Cited by 7 publications

References 52 publications

Entropy rigidity for cusped Hitchin representations

Entropy rigidity for cusped Hitchin representations

Cataclysms for Anosov representations

Cusped Hitchin representations and Anosov representations of geometrically finite Fuchsian groups

Contact Info

Product

Resources

About