On the action of relatively irreducible automorphisms on their train tracks

Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> be a group and let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒢</mml:mi> </mml:math> be a free factor system of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> , namely a free splitting of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>*</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>*</mml:mo> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>*</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>r</mml:mi> </mml:msub> </mml:mrow> </mml:math> . In this paper, we study the set of train track points for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒢</mml:mi> </mml:math> -irreducible automorphisms <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> with exponential growth. Such set is known to coincide with the minimally displaced set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo form="prefix">Min</mml:mo> <mml:mo>(</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> , in the relative deformation space corresponding to the splitting. The theory of such relative spaces, even if it is more general by its own nature, is crucial to understanding reducible automorphisms of free groups, as any automorphism is relatively irreducible with respect to some free factor system <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒢</mml:mi> </mml:math> . Our main result is that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo form="prefix">Min</mml:mo> <mml:mo>(</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is co-compact, under the action of the cyclic subgroup generated by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> . Along the way we obtain other results that could be of independent interest. For instance, we prove that any point of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo form="prefix">Min</mml:mo> <mml:mo>(</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is in uniform distance from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo form="prefix">Min</mml:mo> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>ϕ</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . We also prove that the action of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> on the product of the attracting and the repelling trees for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> , is discrete. Finally, we get some fine insight about the local topology of relative outer space. Some applications of co-compactness are discussed. In particular we generalise a classical result of Bestvina, Feighn and Handel for the centralisers of irreducible automorphisms of free groups, in the more general context of relatively irreducible automorphisms of a free product. From this, we deduce that centralisers of elements of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo form="prefix">Out</mml:mo> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> are finitely generated, which was previously unknown. Finally, we mention that an immediate corollary of co-compactness is that the set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo form="prefix">Min</mml:mo> <mml:mo>(</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is always quasi-isometric to a line.