Entropy rigidity for cusped Hitchin representations

Abstract

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.

Relatively Anosov representations via flows II: Examples

“…In this section, we prove a converse to Proposition 8.6 and characterize the relatively 𝖯 1 -Anosov representations that preserve a properly convex domain. This builds upon work in [17] and extends results in [18,44] from the classical Anosov case to the relative one.…”

“…A projectively visible subgroup acts as a convergence group on its limit set and if, in addition, the action on the limit set is geometrically finite, then the inclusion representation is relatively 𝖯 1 -Anosov. These assertions follow from [17,Prop. 3.5], see Proposition 8.6 below.…”

“…Remark 9.3. The equivalence (2) ⟺ (3) follows from general results in [17] and the implication (2) ⇒ (1) is elementary. So, the new content of Proposition 9.2 is the implication (1) ⇒ (2).…”

“…4.4] implies that 𝜌(Γ) is a 𝖯 𝑘,𝑑−𝑘 -transverse group. Then the equivalence of ( 2) and (3) follows from [17,Prop. 4.4].…”

“…As mentioned in the introduction, a projectively visible subgroup (see Section 1.2.2 for the definition) acts as a convergence group on its limit set [17,Prop. 3.5].…”