Relative hyperbolicity of free extensions of free groups

We study relative hyperbolicity of free-by-free groups: Given a collection of outer automorphisms of the free group we give sufficient conditions for some power of these automorphisms to generate a free group so that the corresponding extension is relatively hyperbolic. Namely, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi 1 comma midline-horizontal-ellipsis phi Subscript k Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> ϕ </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> ⋯ </mml:mo> <mml:msub> <mml:mi> ϕ </mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\phi _1, \cdots \phi _k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a collection of exponentially growing outer automorphisms with a common invariant <italic>subgroup system</italic> such that any conjugacy class in the complement of this system grows exponentially under iteration by all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi Subscript i"> <mml:semantics> <mml:msub> <mml:mi> ϕ </mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\phi _i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then such a subgroup system can be used to construct a collection of peripheral subgroups relative to which, the extension of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {F}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by the free group generated by sufficiently high powers of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi 1 comma midline-horizontal-ellipsis phi Subscript k Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> ϕ </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> ⋯ </mml:mo> <mml:msub> <mml:mi> ϕ </mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\phi _1, \cdots \phi _k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> will be hyperbolic. Moreover, we show that our conditions are also necessary if we are given a free-by-free group with a relative hyperbolic structure where the generating outer automorphisms have <italic>cusp-preserving</italic> property (motivated by homeomorphisms of surfaces which preserve cusps).