Free representations of outer automorphism groups of free products via characteristic abelian coverings

<jats:title>Abstract</jats:title> <jats:p>Given a free product 𝐺, we investigate the existence of faithful free representations of the outer automorphism group <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Out</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2021-0154_ineq_0001.png" /> <jats:tex-math>\operatorname{Out}(G)</jats:tex-math> </jats:alternatives> </jats:inline-formula>, or in other words of embeddings of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Out</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2021-0154_ineq_0001.png" /> <jats:tex-math>\operatorname{Out}(G)</jats:tex-math> </jats:alternatives> </jats:inline-formula> into <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Out</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>F</m:mi> <m:mi>m</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2021-0154_ineq_0003.png" /> <jats:tex-math>\operatorname{Out}(F_{m})</jats:tex-math> </jats:alternatives> </jats:inline-formula> for some 𝑚. This is based on work of Bridson and Vogtmann in which they construct embeddings of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Out</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>F</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2021-0154_ineq_0004.png" /> <jats:tex-math>\operatorname{Out}(F_{n})</jats:tex-math> </jats:alternatives> </jats:inline-formula> into <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Out</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>F</m:mi> <m:mi>m</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2021-0154_ineq_0003.png" /> <jats:tex-math>\operatorname{Out}(F_{m})</jats:tex-math> </jats:alternatives> </jats:inline-formula> for some values of 𝑛 and 𝑚 by interpreting <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Out</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>F</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2021-0154_ineq_0004.png" /> <jats:tex-math>\operatorname{Out}(F_{n})</jats:tex-math> </jats:alternatives> </jats:inline-formula> as the group of homotopy equivalences of a graph 𝑋 of genus 𝑛, and by lifting homotopy equivalences of 𝑋 to a characteristic abelian cover of genus 𝑚. Our construction for a free product 𝐺, using a presentation of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Out</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2021-0154_ineq_0001.png" /> <jats:tex-math>\operatorname{Out}(G)</jats:tex-math> </jats:alternatives> </jats:inline-formula> due to Fuchs-Rabinovich, is written as an algebraic proof, but it is directly inspired by Bridson and Vogtmann’s topological method and can be interpreted as lifting homotopy equivalences of a graph of groups. For instance, we obtain a faithful free representation of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Out</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2021-0154_ineq_0001.png" /> <jats:tex-math>\operatorname{Out}(G)</jats:tex-math> </jats:alternatives> </jats:inline-formula> when <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>F</m:mi> <m:mi>d</m:mi> </m:msub> <m:mo>∗</m:mo> <m:msub> <m:mi>G</m:mi> <m:mrow> <m:mi>d</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>∗</m:mo> <m:mi mathvariant="normal">⋯</m:mi> <m:mo>∗</m:mo> <m:msub> <m:mi>G</m:mi> <m:mi>n</m:mi> </m:msub> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2021-0154_ineq_0009.png" /> <jats:tex-math>G=F_{d}\ast G_{d+1}\ast\cdots\ast G_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, with <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>F</m:mi> <m:mi>d</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2021-0154_ineq_0010.png" /> <jats:tex-math>F_{d}</jats:tex-math> </jats:alternatives> </jats:inline-formula> free of rank 𝑑 and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>G</m:mi> <m:mi>i</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2021-0154_ineq_0011.png" /> <jats:tex-math>G_{i}</jats:tex-math> </jats:alternatives> </jats:inline-formula> finite abelian of order coprime to <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2021-0154_ineq_0012.png" /> <jats:tex-math>n-1</jats:tex-math> </jats:alternatives> </jats:inline-formula>.</jats:p> / ENS Lyon studentship