A collar lemma for partially hyperconvex surface group representations

Abstract

We show that a collar lemma holds for Anosov representations of fundamental groups of surfaces into SL(n, 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-positive representations into SO(p, q) if p ≥ 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. Contents1. Introduction 2 2. Preliminaries 7 2.1. Anosov representations 8 3. Cross ratios 12 3.1. Projective cross ratios 12 3.2. Grassmannian cross ratio 13 3.3. Relations of the cross ratios 15 4. Partial hyperconvexity 15 4.1. Property H k 15 4.2. Property C k 18 4.3. Reducible representations and Fuchsian locii 22 4.4. Θ-positive representations 23 5. Positively ratioed representations 26 6. Proof of the collar lemma 28 7. A counterexample to the strong collar lemma 36 References 39

Positive surface group representations in PO(p,q)

“…In higher rank many different measures of the magnitude of an element play an important role in understanding geometric features of actions, and generalizing different properties of hyperbolizations: the fundamental weights ω k describe the translation length on the symmetric space with respect to suitable Finsler distances [KLP17], and, as mentioned above, behave like the hyperbolic length function under surgery for representations in higher rank Teichmüller spaces; despite the roots α k are not induced by a distance, for Anosov representations they are coarsly equivalent to the stable length with respect to any generating system [KP20], and, as for Teichmüller space, their entropy is constant and equal to one on higher rank Teichmüller spaces [PS17,PSW19]. Theorem B encodes a powerful generalization of another feature of holonomies of hyperbolizations to Θ-positive representations into PO(p, q); analogue results were previously established for Hitchin representations [LZ17], Maximal representations [BP17] and representations that satisfy some partial hyperconvexity properties [BP20].…”

“…If k < p − 1, we proved in [BP20] that Equation (12) follows from property H k . In the case of k = p − 1 we will reduce the claim to an inequality for the PO(2, q)positive structure, which we prove in Section 5.1.…”

“…Theorem B follows from the results of [BP20] for k < p − 2, and is new for k = p − 2, p − 1. To deal with the two additional cases we need a different approach compared to [BP20] since Θ positive representation are in general not p-Anosov and don't satisfy additional transversality properties. There are two key tools, building on the positive structure, that allow us to circumvent this problem.…”

“…It was shown in [BP20] that positive representations into PO(p, q) are k-positively ratioed for k ≤ p − 2; thus Theorem A is only new for k = p − 1. We include a new proof of the other cases as well, since it is better adapted to the positive structure and builds on two results of independent interest which we also need for the proof of the collar lemmas: First we show that any positive triple (x, y, z) naturally defines tangent cones c + k (x) in T x Iso k (R p,q ), and the boundary map of a positive representation containing (x, y, z) is tangent to c + k (x) (Proposition 4.4, see also Remark 4.6).…”

“…With Proposition 5.6 at hand we can reduce the proof of Theorem B to an inequality for the positive structure for PO(2, q), which we establish in Proposition 5.5. As it does not cost any extra effort we include proofs of the collar lemmas for the cases k < p − 2 as well, which are simpler and more direct than the ones in [BP20], making this work independent from [BP20].…”