The small index property for free nilpotent groups

2011/11/23 by Vladimir Tolstykh, Tolstykh, Vladimir · 3 citations
Mathematics · #20F19 (Secondary) #20F28 (Primary) 20E05 #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO) #Rings, Modules, and Algebras #math.GR #math.LO #msc:20E05 #msc:20F19 #msc:20F28

paper · pdf · doi:10.48550/arxiv.1111.5635

18 pp

openalex publication_date 2011/11/23 · arxiv created 2012/04/25 · arxiv updated 2012/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let F be a relatively free algebra of infinite rank. We say that F has the SMALL INDEX PROPERTY if any subgroup of Gamma=Aut(F) of index at most rank(F) contains the pointwise stabilizer Gamma_(U) of a subset U of F of cardinality less than rank(F). We prove that every infinitely generated free nilpotent/abelian group has the small index property, and discuss a number of applications.

Cited by

• Principal congruence subgroups in the infinite rank case
• On the outer automorphism groups of free groups
• Complete outer automorphism groups of free nilpotent groups

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