an:01485627
Zbl 0984.20025
Bestvina, Mladen; Feighn, Mark; Handel, Michael
The Tits alternative for \(\text{Out}(F_n)\). I: Dynamics of exponentially-growing automorphisms
EN
Ann. Math. (2) 151, No. 2, 517-623 (2000).
00064445
2000
j
20F28 57M07 20E36 20E07 20E05
train tracks; attracting laminations; Tits alternative; outer automorphism groups; free groups; free subgroups; subgroups of finite index
A group satisfies the Tits alternative if each of its subgroups either contains a free subgroup of rank two or is virtually solvable. The Tits alternative is satisfied by finitely generated linear groups [\textit{J. Tits}, J. Algebra 20, 250-270 (1972; Zbl 0236.20032)] and mapping class groups of surfaces [\textit{N. V. Ivanov}, Dokl. Akad. Nauk SSSR 275, 786-789 (1984; Zbl 0586.20026), \textit{J. McCarthy}, Trans. Am. Math. Soc. 291, 583-612 (1985; Zbl 0579.57006)].
In a series of two papers the authors prove that the outer automorphism group \(\text{Out}(F_n)\) of a free group of rank \(n\) satisfies the Tits alternative. In a third paper the authors show that a solvable subgroup of \(\text{Out}(F_n)\) has a finitely generated free Abelian subgroup of index at most \(3^{5n^2}\).
In this first paper the authors outline the contents of all three papers and provide the general framework for the subject (``relative train tracks'', ``attracting laminations''). The main theorem of this paper is the following: Suppose that \(H\) is a subgroup of \(\text{Out}(F_n)\) that does not contain a free subgroup of rank 2. Then there is a finite index subgroup \(H_0\) of \(H\), a finitely generated free Abelian group \(A\), and a map \(\Phi\colon H_0\to A\) such that every element of \(\text{Ker}(\Phi)\) has polynomial growth and unipotent image in \(\text{GL}(n,\mathbb{Z})\).
Wolfgang Heil (Tallahassee)
Zbl 0236.20032; Zbl 0586.20026; Zbl 0579.57006