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Abelian subgroups of $$\text{Out}(F_n)$$. (English) Zbl 1201.20031
In this long paper the authors classify up to finite index the Abelian subgroups of $$\text{Out}(F_n)$$, the group of outer automorphisms of the free group $$F_n$$ of rank $$n$$. First they construct an Abelian subgroup $$\mathcal D(\varphi)$$ from a given $$\varphi\in\text{Out}(F_n)$$ by a process called disintegration.
Then they prove the following (main) Theorem: For every Abelian subgroup $$A$$ of $$\text{Out}(F_n)$$ there exists $$\varphi\in A$$ such that $$A\cap\mathcal D(\varphi)$$ has finite index in $$A$$.
By applying this theorem they give an explicit description, in terms of relative train track maps and up to finite index, of all maximal rank Abelian subgroups of $$\text{Out}(F_n)$$ and $$\text{IA}_n$$.

##### MSC:
 20F28 Automorphism groups of groups 20E07 Subgroup theorems; subgroup growth 20E05 Free nonabelian groups 20F05 Generators, relations, and presentations of groups
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##### References:
 [1] H Bass, A Lubotzky, Linear-central filtrations on groups (editors W Abikoff, J S Birman, K Kuiken), Contemp. Math. 169, Amer. Math. Soc. (1994) 45 · Zbl 0817.20038 [2] M Bestvina, M Feighn, M Handel, The Tits alternative for $$\mathrm{Out}(F_n)$$. I. Dynamics of exponentially-growing automorphisms, Ann. of Math. $$(2)$$ 151 (2000) 517 · Zbl 0984.20025 · doi:10.2307/121043 · www.math.princeton.edu · eudml:121772 [3] M Bestvina, M Feighn, M Handel, Solvable subgroups of $$\mathrm{Out}(F_n)$$ are virtually Abelian, Geom. Dedicata 104 (2004) 71 · Zbl 1052.20027 · doi:10.1023/B:GEOM.0000022864.30278.34 [4] M Bestvina, M Handel, Train tracks and automorphisms of free groups, Ann. of Math. $$(2)$$ 135 (1992) 1 · Zbl 0757.57004 · doi:10.2307/2946562 [5] D Cooper, Automorphisms of free groups have finitely generated fixed point sets, J. Algebra 111 (1987) 453 · Zbl 0628.20029 · doi:10.1016/0021-8693(87)90229-8 [6] M Culler, Finite groups of outer automorphisms of a free group (editors K I Appel, J G Ratcliffe, P E Schupp), Contemp. Math. 33, Amer. Math. Soc. (1984) 197 · Zbl 0552.20024 [7] M Culler, K Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986) 91 · Zbl 0589.20022 · doi:10.1007/BF01388734 · eudml:143335 [8] B Farb, M Handel, Commensurations of $$\mathrm{Out}(\mathrmF_n)$$, Publ. Math. Inst. Hautes Études Sci. (2007) 1 · Zbl 1137.20018 · doi:10.1007/s10240-007-0007-7 · numdam:PMIHES_2007__105__1_0 · eudml:104223 [9] A Fathi, L Laudenbach, V Poénaru, editors, Travaux de Thurston sur les surfaces, Astérisque 66, Soc. Math. France (1979) 284 [10] M. Feighn, M Handel, The recognition theorem for $$\Out(F_n)$$, Preprint · Zbl 1239.20036 · doi:10.4171/GGD/116 [11] J Franks, M Handel, K Parwani, Fixed points of abelian actions, J. Mod. Dyn. 1 (2007) 443 · Zbl 1130.37023 · doi:10.3934/jmd.2007.1.443 [12] J Franks, M Handel, K Parwani, Fixed points of abelian actions on $$S^2$$, Ergodic Theory Dynam. Systems 27 (2007) 1557 · Zbl 1143.37030 · doi:10.1017/S0143385706001088 [13] D Gaboriau, A Jaeger, G Levitt, M Lustig, An index for counting fixed points of automorphisms of free groups, Duke Math. J. 93 (1998) 425 · Zbl 0946.20010 · doi:10.1215/S0012-7094-98-09314-0 [14] G Levitt, M Lustig, Most automorphisms of a hyperbolic group have very simple dynamics, Ann. Sci. École Norm. Sup. $$(4)$$ 33 (2000) 507 · Zbl 0997.20043 · doi:10.1016/S0012-9593(00)00120-8 · numdam:ASENS_2000_4_33_4_507_0 · eudml:82525 [15] G Levitt, M Lustig, Automorphisms of free groups have asymptotically periodic dynamics, J. Reine Angew. Math. 619 (2008) 1 · Zbl 1157.20017 · doi:10.1515/CRELLE.2008.038 [16] J Nielsen, Die Isomorphismengruppe der freien Gruppen, Math. Ann. 91 (1924) 169 · JFM 50.0078.04 · doi:10.1007/BF01556078 [17] W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. $$($$N.S.$$)$$ 19 (1988) 417 · Zbl 0674.57008 · doi:10.1090/S0273-0979-1988-15685-6
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