Propriétés résiduelles des groupes, et méthodes probabilistes (Q2748283)

The authors announce the following results. The proofs, which are based on probabilistic methods, will be published elsewhere. Theorem 1: If \(S\) is a finite nonabelian simple group and \(w(X,Y)\) is a nontrivial element of the free group \(F_2\) on \(X,Y\), then the probability that two random elements \(x,y\in S\) satisfy \(w(x,y)\neq 1\) tends to \(1\) as \(|S|\to\infty\). Corollary 2 (T. Weigel): \(F_2\) is residually-\(\mathcal C\) for any infinite set \(\mathcal C\) of finite nonabelian simple groups. Theorem 3: \(\text{PSL}_2(\mathbb{Z})\) is residually-\(\mathcal C\) for any infinite set of finite nonabelian simple groups not containing \(\text{PSp}_4(q)\) (\(q\) a power of \(2\) or \(3\)) or \(\text{Sz}(q)\). Theorem 4: If \(A\) and \(B\) are finite nontrivial groups which are not both \(2\)-groups, then the free product \(A*B\) is residually-\(\mathcal C\) for any set \(\mathcal C\) of finite simple classical groups of unbounded rank. Corollary 5: If \(d\geq 3\), then the profinite completion of \(\text{SL}_d(\mathbb{Z})\) has a dense free subgroup of finite rank. Theorem 6: If \(G\) is the profinite completion of a finitely generated linear group \(\Gamma\) over some field \(K\), then either \(\Gamma\) is virtually solvable or \(G\) has an open subgroup containing a dense free subgroup of finite rank.