Algebraic extensions in free groups

Alexei Miasnikov, Enric Ventura, Pascal Weil

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

37 Scopus citations

Abstract

The aim of this paper is to unify the points of view of three recent and independent papers (Ventura 1997, Margolis, Sapir and Weil 2001 and Kapovich and Miasnikov 2002), where similar modern versions of a 1951 theorem of Takahasi were given. We develop a theory of algebraic extensions for free groups, highlighting the analogies and differences with respect to the corresponding classical field-theoretic notions, and we discuss in detail the notion of algebraic closure. We apply that theory to the study and the computation of certain algebraic properties of subgroups (e.g., being malnormal, pure, inert or compressed, being closed in certain profinite topologies) and the corresponding closure operators. We also analyze the closure of a subgroup under the addition of solutions of certain sets of equations.

Original languageEnglish
Title of host publicationGeometric Group Theory
EditorsGoulnara N. Arzhantseva, Laurent Bartholdi, José Burillo, Enric Ventura
Pages225-253
Number of pages29
DOIs
StatePublished - 2007
EventInternational Conferences on Asymptotic and Probabilistic Methods in Geometric Group Theory, 2005 - Geneva, Switzerland
Duration: 20 Jun 200525 Jun 2005

Publication series

NameTrends in Mathematics
Volume40
ISSN (Print)2297-0215
ISSN (Electronic)2297-024X

Conference

ConferenceInternational Conferences on Asymptotic and Probabilistic Methods in Geometric Group Theory, 2005
Country/TerritorySwitzerland
CityGeneva
Period20/06/0525/06/05

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