## THE ACTION OF THE MAPPING CLASS GROUP ON MAXIMAL REPRESENTATIONS

Citations: | 3 - 0 self |

### BibTeX

@MISC{Wienhard_theaction,

author = {Anna Wienhard},

title = {THE ACTION OF THE MAPPING CLASS GROUP ON MAXIMAL REPRESENTATIONS},

year = {}

}

### OpenURL

### Abstract

Abstract. Let Γg be the fundamental group of a closed oriented Riemann surface Σg, g ≥ 2, and let G be a simple Lie group of Hermitian type. The Toledo invariant defines the subset of maximal representations Repmax(Γg, G) in the representation variety Rep(Γg, G). Repmax(Γg, G) is a union of connected components with similar properties as Teichmüller space T (Σg) = Repmax(Γg, PSL(2, R)). We prove that the mapping class group ModΣg acts properly on Repmax(Γg, G) when G = Sp(2n, R), SU(n, n), SO ∗ (4n), Spin(2, n).

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Citation Context ...lued function T : Rep(Γg, G) → Z called the Toledo invariant. The level set of the maximal possible modulus of T is the set of maximal representations Rep max(Γg, G) ⊂ Rep(Γg, G), which is studied in =-=[12, 13, 23, 15, 1, 14, 6, 3, 4, 21]-=-. Since the Toledo invariant is locally constant, its level sets are unions of connected components. Results of [12, 13, 6, 4] suggest that maximal representations provide a meaningful generalization ... |

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Citation Context ...→ ρ(ψ −1 γ) )] . Recall that Out(Γg) is isomorphic to π0(Diff(Σg)). The mapping class group ModΣg is the subgroup of Out(Γg) corresponding to orientation preserving diffeomorphisms of Σg. We refer to =-=[16, 10]-=- for a general introduction to mapping class groups and to [11] for a recent survey on dynamical properties of the action of Out(Γg) on representation varieties Rep(Γg, G). This note is concerned with... |

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Citation Context ...→ ρ(ψ −1 γ) )] . Recall that Out(Γg) is isomorphic to π0(Diff(Σg)). The mapping class group ModΣg is the subgroup of Out(Γg) corresponding to orientation preserving diffeomorphisms of Σg. We refer to =-=[16, 10]-=- for a general introduction to mapping class groups and to [11] for a recent survey on dynamical properties of the action of Out(Γg) on representation varieties Rep(Γg, G). This note is concerned with... |

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Citation Context ...lued function T : Rep(Γg, G) → Z called the Toledo invariant. The level set of the maximal possible modulus of T is the set of maximal representations Rep max(Γg, G) ⊂ Rep(Γg, G), which is studied in =-=[12, 13, 23, 15, 1, 14, 6, 3, 4, 21]-=-. Since the Toledo invariant is locally constant, its level sets are unions of connected components. Results of [12, 13, 6, 4] suggest that maximal representations provide a meaningful generalization ... |

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Citation Context ...lued function T : Rep(Γg, G) → Z called the Toledo invariant. The level set of the maximal possible modulus of T is the set of maximal representations Rep max(Γg, G) ⊂ Rep(Γg, G), which is studied in =-=[12, 13, 23, 15, 1, 14, 6, 3, 4, 21]-=-. Since the Toledo invariant is locally constant, its level sets are unions of connected components. Results of [12, 13, 6, 4] suggest that maximal representations provide a meaningful generalization ... |

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Citation Context ...udy of the Hitchin component Rep H(Γg, G) ⊂ Rep(Γg, G) for split real simple Lie groups G. François Labourie recently announced, as a consequence of his work on Anosov representations and crossratios =-=[18, 20]-=-, that the mapping class group acts properly on Rep H(Γg, SL(n, R)). After finishing this note, we learned that he also has a proof for maximal representations into Sp(2n, R) [19]. The author is indeb... |

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Citation Context ... The mapping class group ModΣg is the subgroup of Out(Γg) corresponding to orientation preserving diffeomorphisms of Σg. We refer to [16, 10] for a general introduction to mapping class groups and to =-=[11]-=- for a recent survey on dynamical properties of the action of Out(Γg) on representation varieties Rep(Γg, G). This note is concerned with the action of the mapping class group on special connected com... |

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Citation Context ...m(Γg, G) → Z. The map T is invariant under the action of Inn(G) and constant on connected components of the representation variety. The Toledo invariant satisfies a generalized Milnor-Wood inequality =-=[8, 7]-=- |T| ≤ pX rkX |χ(Σg)|, 2 where rkX is the real rank of X and pX ∈ N is explicitly computable in terms of the root system. Definition 2.1. A representation ρ : Γg → G is said to be maximal if |T(ρ)| = ... |

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Citation Context ...m(Γg, G) → Z. The map T is invariant under the action of Inn(G) and constant on connected components of the representation variety. The Toledo invariant satisfies a generalized Milnor-Wood inequality =-=[8, 7]-=- |T| ≤ pX rkX |χ(Σg)|, 2 where rkX is the real rank of X and pX ∈ N is explicitly computable in terms of the root system. Definition 2.1. A representation ρ : Γg → G is said to be maximal if |T(ρ)| = ... |

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Citation Context ...ROUP 7 Corollary 3.2. The action of ModΣg proper. on Rep max(Γg, Sp(2n, R)) is That Theorem 1.1 can be deduced from Proposition 3.1 and Proposition 2.4 can be seen as follows - we refer the reader to =-=[3, 5, 24]-=- for more on tight homomorphisms and their properties. Satake [22, Ch. IV] investigated when a simple Lie group G of Hermitian type admits a homomorphism τ : G → Sp(2m, R). such that the induced homom... |

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Citation Context ...udy of the Hitchin component Rep H(Γg, G) ⊂ Rep(Γg, G) for split real simple Lie groups G. François Labourie recently announced, as a consequence of his work on Anosov representations and crossratios =-=[18, 20]-=-, that the mapping class group acts properly on Rep H(Γg, SL(n, R)). After finishing this note, we learned that he also has a proof for maximal representations into Sp(2n, R) [19]. The author is indeb... |

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Citation Context ...entations and Hitchin components are always contractibleACTION OF MAPPING CLASS GROUP 5 subsets of Rep(Γg, G), certain components of the set of maximal representations might have nontrivial topology =-=[14, 2]-=-. 2.2. Translation Lengths. For a hyperbolization h : Γg → PSL(2, R) define the translation length trh of γ ∈ Γg as trh(γ) := inf p∈ d (p, γp). For a representation ρ : Γg → G define similarly the tra... |

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Citation Context ... and crossratios [18, 20], that the mapping class group acts properly on Rep H(Γg, SL(n, R)). After finishing this note, we learned that he also has a proof for maximal representations into Sp(2n, R) =-=[19]-=-. The author is indebted to Marc Burger for motivation, interesting discussions and for pointing out a mistake in a preliminary version of this paper. The author thanks Bill Goldman, Ursula Hamenstädt... |

1 |
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