Obstructions for Minimal Distal Actions

Abstract

In this paper, we mainly consider the nonexistences of minimal distal actions by some groups on compact manifolds, particularly on surfaces. Suppose that X is a compact manifold and Γ is a finitely generated group acting on X. We show in the following cases that Γ cannot act on X minimally and distally. (1) X is connected and the first Čech cohomology group Ȟ¹(X) with integer coefficients is nontrivial and Γ is amenable; (2) X is the 2-sphere \(\mathbb{S}^{2}\) or the real projective plane ℝℙ² and Γ contains no nonabelian free subgroup; (3) X is a closed surface and Γ is a lattice of SL(n, ℝ)(n ≥ 3).