English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Given a free product , we investigate the existence of faithful free representations of the outer automorphism groupOut⁡(G)\operatorname{Out}(G), or in other words of embeddings ofOut⁡(G)\operatorname{Out}(G)intoOut⁡(Fm)\operatorname{Out}(F_{m})for some . This is based on work of Bridson and Vogtmann in which they construct embeddings ofOut⁡(Fn)\operatorname{Out}(F_{n})intoOut⁡(Fm)\operatorname{Out}(F_{m})for some values of  and  by interpretingOut⁡(Fn)\operatorname{Out}(F_{n})as the group of homotopy equivalences of a graph  of genus , and by lifting homotopy equivalences of  to a characteristic abelian cover of genus . Our construction for a free product , using a presentation ofOut⁡(G)\operatorname{Out}(G)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 ofOut⁡(G)\operatorname{Out}(G)whenG=Fd∗Gd+1∗⋯∗GnG=F_{d}\ast G_{d+1}\ast\cdots\ast G_{n}, withFdF_{d}free of rank  andGiG_{i}finite abelian of order coprime ton-1n-1.

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