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<p>In this paper, we mainly consider the nonexistences of minimal distal actions by some groups on compact manifolds, particularly on surfaces. Suppose that <i>X</i> is a compact manifold and Γ is a finitely generated group acting on <i>X</i>. We show in the following cases that Γ cannot act on <i>X</i> minimally and distally. (1) <i>X</i> is connected and the first Čech cohomology group <i>Ȟ</i><sup>1</sup>(<i>X</i>) with integer coefficients is nontrivial and Γ is amenable; (2) <i>X</i> is the 2-sphere <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb{S}^{2}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> or the real projective plane ℝℙ<sup>2</sup> and Γ contains no nonabelian free subgroup; (3) <i>X</i> is a closed surface and Γ is a lattice of SL(<i>n</i>, ℝ)(<i>n</i> ≥ 3).</p>

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Obstructions for Minimal Distal Actions

  • 期刊论文

  • 发表日期：

    2026年2月15日

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  • Enhui Shi,
  • Hui Xu,
  • Lizhen Zhou

摘要

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 \(\mathbb{S}^{2}\) 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).