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<p>Let <i>F</i> be a free group of positive, finite rank, and let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\phi :F \rightarrow F\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϕ</mi> <mo>:</mo> <mi>F</mi> <mo stretchy="false">→</mo> <mi>F</mi> </mrow> </math></EquationSource> </InlineEquation> be an automorphism. In this paper, we study some geometric aspects of the free-by-cyclic groups <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(F \rtimes _{\phi } \mathbb {Z}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <msub> <mo>⋊</mo> <mi>ϕ</mi> </msub> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation>. We provide a positive solution to a conjecture proposed by Gersten in (Geom. Funct. Anal. 4: 37–51, 1994). Furthermore, we show that for a large class of free-by-cyclic groups, two notions of finite height and strongly quasiconvex are equivalent. This partially addresses a question posed by Nguyen-Tran-Yang in (Math. Ann. 381: 405–437, 2021).</p>

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Some Remarks on Subgroups of Free-by-Cyclic Groups

  • 期刊论文

  • 发表日期：

    2025年7月26日

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  • Nhat Minh Doan,
  • Hoang Thanh Nguyen

摘要

Let F be a free group of positive, finite rank, and let \(\phi :F \rightarrow F\) ϕ : F → F be an automorphism. In this paper, we study some geometric aspects of the free-by-cyclic groups \(F \rtimes _{\phi } \mathbb {Z}\) F ⋊ ϕ Z . We provide a positive solution to a conjecture proposed by Gersten in (Geom. Funct. Anal. 4: 37–51, 1994). Furthermore, we show that for a large class of free-by-cyclic groups, two notions of finite height and strongly quasiconvex are equivalent. This partially addresses a question posed by Nguyen-Tran-Yang in (Math. Ann. 381: 405–437, 2021).