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

DOI10.1515/JGTH-2021-0154arXiv2107.11230MaRDI QIDQ6373585

Publication date: 23 July 2021

Abstract: Given a free product

G

, we investigate the existence of faithful free representations of the outer automorphism group

e x t O u t (G)

, or in other words of embeddings of

e x t O u t (G)

into

e x t O u t l e f t (F_{m} i g h t)

for some

m

. This is based on a work of Bridson and Vogtmann in which they construct embeddings of

e x t O u t l e f t (F_{n} i g h t)

into

e x t O u t l e f t (F_{m} i g h t)

for some values of

n

and

m

by interpreting

e x t O u t l e f t (F_{n} i g h t)

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

e x t O u t (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 of

e x t O u t (G)

when

G = F_{d} a s t G_{d + 1} a s t c d o t s a s t G_{n}

, with

F_{d}

free of rank

d

and

G_{i}

finite abelian of order coprime to

n - 1

.

Mathematics Subject Classification ID

Automorphisms of infinite groups (20E36) Automorphism groups of groups (20F28) Free nonabelian groups (20E05) Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations (20E06)

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