TY - JOUR
T1 - A Classification of Fully Residually Free Groups of Rank Three or Less
AU - Fine, Benjamin
AU - Gaglione, Anthony M.
AU - Myasnikov, Alexei
AU - Rosenberger, Gerhard
AU - Spellman, Dennis
PY - 1998/2/15
Y1 - 1998/2/15
N2 - A groupGisfully residually freeprovided to every finite setS⊂G\{1} of non-trivial elements ofGthere is a free groupFSand an epimorphismhS:G→FSsuch thathS(g)≠1 for allg∈S. Ifnis a positive integer, then a groupGisn-freeprovided every subgroup ofGgenerated bynor fewer distinct elements is free. Our main result shows that a fully residually free group of rank at most 3 is either abelian, free, or a free rank one extension of centralizers of a rank two free group. To prove this we prove that every 2-free, fully residually free group is actually 3-free. There are fully residually free groups which are not 2-free and there are 3-free, fully residually free groups which are not 4-free.
AB - A groupGisfully residually freeprovided to every finite setS⊂G\{1} of non-trivial elements ofGthere is a free groupFSand an epimorphismhS:G→FSsuch thathS(g)≠1 for allg∈S. Ifnis a positive integer, then a groupGisn-freeprovided every subgroup ofGgenerated bynor fewer distinct elements is free. Our main result shows that a fully residually free group of rank at most 3 is either abelian, free, or a free rank one extension of centralizers of a rank two free group. To prove this we prove that every 2-free, fully residually free group is actually 3-free. There are fully residually free groups which are not 2-free and there are 3-free, fully residually free groups which are not 4-free.
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U2 - 10.1006/jabr.1997.7205
DO - 10.1006/jabr.1997.7205
M3 - Article
AN - SCOPUS:0000706653
SN - 0021-8693
VL - 200
SP - 571
EP - 605
JO - Journal of Algebra
JF - Journal of Algebra
IS - 2
ER -