Cross Ratios on Boundaries of Symmetric Spaces and Euclidean Buildings

Beyrer, Jonas

doi:10.1007/s00031-020-09549-5

Cited by 4 publications

(1 citation statement)

References 35 publications

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“…There is a natural R-valued cross ratio on a subset of G P × G P − × G P − × G P for any (semi-)simple G, P < G a maximal parabolic and P − a opposite parabolic (e.g. [Bey21]). Condition (Tr) can be defined analogously, and Proposition can be proved in the same way in this more general setting.…”

Section: 2mentioning

confidence: 99%

Degenerations of k-positive surface group representations

Beyrer,

Pozzetti

2021

Preprint

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We introduce k-positive representations, a large class of {1, . . . , k}-Anosov surface group representations into PGL(E) that share many features with Hitchin representations, and we study their degenerations: unless they are Hitchin, they can be deformed to non-discrete representations, but any limit is at least (k − 3)-positive and irreducible limits are (k − 1)-positive. A major ingredient, of independent interest, is a general limit theorem for positively ratioed representations. JONAS BEYRER AND BEATRICE POZZETTI 6.2. k-positive representations 27 6.3. Transversality properties of k-positive representations 28 6.4. k-tridirect representations 30 7. Degenerations of k-positive representations 31 7.1. Anosov limits of k-positive representations 31 7.2. Limits of k-positive representations 32 7.3. Coherence of the boundary map in the limit 33 8. Examples and open questions 36 8.1. All transversality could get lost in the limit 37 8.2. The set of positively ratioed representations is not open 37 8.3. Limits of k-positive representations 38 References 39

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Section: 2mentioning

confidence: 99%

Degenerations of k-positive surface group representations

Beyrer,

Pozzetti

2021

Preprint

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A collar lemma for partially hyperconvex surface group representations

Beyrer

Pozzetti

2021

Trans. Amer. Math. Soc.

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We show that a collar lemma holds for Anosov representations of fundamental groups of surfaces into S L ( n , R ) SL(n,\mathbb {R}) that satisfy partial hyperconvexity properties inspired from Labourie’s work. This is the case for several open sets of Anosov representations not contained in higher rank Teichmüller spaces, as well as for Θ \Theta -positive representations into S O ( p , q ) SO(p,q) if p ≥ 4 p\geq 4 . We moreover show that ‘positivity properties’ known for Hitchin representations, such as being positively ratioed and having positive eigenvalue ratios, also hold for partially hyperconvex representations.

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Patterson–Sullivan theory for Anosov subgroups

Dey

Kapovich

2022

Trans. Amer. Math. Soc.

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We extend several notions and results from the classical Patterson–Sullivan theory to the setting of Anosov subgroups of higher rank semisimple Lie groups, working primarily with invariant Finsler metrics on associated symmetric spaces. In particular, we prove the equality between the Hausdorff dimensions of flag limit sets, computed with respect to a suitable Gromov (pre-)metric on the flag manifold, and the Finsler critical exponents of Anosov subgroups.

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Cross Ratios on Boundaries of Symmetric Spaces and Euclidean Buildings

Cited by 4 publications

References 35 publications

Degenerations of k-positive surface group representations

Degenerations of k-positive surface group representations

A collar lemma for partially hyperconvex surface group representations

Patterson–Sullivan theory for Anosov subgroups

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