TY - JOUR
T1 - Residual properties of free groups and probabilistic methods
AU - Dixon, John D.
AU - Pyber, László
AU - Seress, Ákos
AU - Shalev, Aner
PY - 2003
Y1 - 2003
N2 - Let w be a non-trivial word in two variables. We prove that the probability that two randomly chosen elements x, y of a nonabelian finite simple group S satisfy w(x, y) = 1 tends to 0 as |S| → ∞. As a consequence of this result, we obtain a new short proof of a well-known conjecture of Magnus concerning free groups. We also use it to prove an analogue of the Tits alternative: If a linear group Γ is not virtually solvable then its profinite completion Γ̂ has a "virtually dense" free subgroup of finite rank.
AB - Let w be a non-trivial word in two variables. We prove that the probability that two randomly chosen elements x, y of a nonabelian finite simple group S satisfy w(x, y) = 1 tends to 0 as |S| → ∞. As a consequence of this result, we obtain a new short proof of a well-known conjecture of Magnus concerning free groups. We also use it to prove an analogue of the Tits alternative: If a linear group Γ is not virtually solvable then its profinite completion Γ̂ has a "virtually dense" free subgroup of finite rank.
UR - http://www.scopus.com/inward/record.url?scp=0037248840&partnerID=8YFLogxK
U2 - 10.1515/crll.2003.019
DO - 10.1515/crll.2003.019
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AN - SCOPUS:0037248840
SN - 0075-4102
SP - 159
EP - 172
JO - Journal fur die Reine und Angewandte Mathematik
JF - Journal fur die Reine und Angewandte Mathematik
IS - 556
ER -