Entropy and affine actions for surface groups

Abstract

We give a short and independent proof of a theorem of Danciger and Zhang: surface groups with Hitchin linear part cannot act properly on the affine space. The proof is fundamentally different and relies on ergodic methods.

Affine Anosov Representations and Proper Actions

“…Subsequently, Labourie [Lab01] showed that if Γ is the fundamental group of a compact orientable surface without boundary and of genus atleast two and (ρ, u) : Γ → SO n+1,n ⋉ R 2n+1 is such that ρ is Fuchsian, then (ρ, u)(Γ) does not act properly on R 2n+1 . Recently, Danciger-Zhang [DZ19] and Labourie [Lab22] generalized Labourie's result and showed that if Γ is the fundamental group of a compact orientable surface without boundary and of genus atleast two and (ρ, u) : Γ → SO n+1,n ⋉ R 2n+1 is such that ρ is Hitchin, then (ρ, u)(Γ) does not act properly on R 2n+1 . Both these results generalize the first reformulation of Mess' result.…”