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<p>Let <i>G</i> be a connected semisimple real algebraic group. The class of transverse subgroups of <i>G</i> includes all discrete subgroups of rank one Lie groups and any subgroups of Anosov or relative Anosov subgroups. Given a transverse subgroup <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3292_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>, we show that the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3292_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-action on the Weyl chamber flow space determined by its limit set is properly discontinuous. This allows us to consider the quotient space and define Bowen–Margulis–Sullivan measures. We then establish the ergodic dichotomy for the Weyl chamber flow, in the original spirit of Hopf–Tsuji–Sullivan. We also introduce the notion of growth indicators and discuss their properties and roles in the study of conformal measures, extending the work of Quint. We discuss several applications as well.</p>

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Properly discontinuous actions, growth indicators, and conformal measures for transverse subgroups

  • 期刊论文

  • 发表日期：

    2025年9月23日

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  • Dongryul M. Kim,
  • Hee Oh,
  • Yahui Wang

摘要

Let G be a connected semisimple real algebraic group. The class of transverse subgroups of G includes all discrete subgroups of rank one Lie groups and any subgroups of Anosov or relative Anosov subgroups. Given a transverse subgroup \(\Gamma \) Γ , we show that the \(\Gamma \) Γ -action on the Weyl chamber flow space determined by its limit set is properly discontinuous. This allows us to consider the quotient space and define Bowen–Margulis–Sullivan measures. We then establish the ergodic dichotomy for the Weyl chamber flow, in the original spirit of Hopf–Tsuji–Sullivan. We also introduce the notion of growth indicators and discuss their properties and roles in the study of conformal measures, extending the work of Quint. We discuss several applications as well.