TY - GEN
T1 - Algebraic extensions in free groups
AU - Miasnikov, Alexei
AU - Ventura, Enric
AU - Weil, Pascal
N1 - Publisher Copyright:
© 2007 Birkhäuser Verlag Basel/Switzerland.
PY - 2007
Y1 - 2007
N2 - 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.
AB - 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.
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U2 - 10.1007/978-3-7643-8412-8_12
DO - 10.1007/978-3-7643-8412-8_12
M3 - Conference contribution
AN - SCOPUS:84924815109
SN - 9783764384111
T3 - Trends in Mathematics
SP - 225
EP - 253
BT - Geometric Group Theory
A2 - Arzhantseva, Goulnara N.
A2 - Bartholdi, Laurent
A2 - Burillo, José
A2 - Ventura, Enric
T2 - International Conferences on Asymptotic and Probabilistic Methods in Geometric Group Theory, 2005
Y2 - 20 June 2005 through 25 June 2005
ER -