Kapovich, Ilya Algorithmic detectability of iwip automorphisms. (English) Zbl 1319.20030 Bull. Lond. Math. Soc. 46, No. 2, 279-290 (2014). From the introduction: The notion of a pseudo-Anosov homeomorphism of a compact surface plays a fundamental role in low-dimensional topology and the study of mapping class groups. In the context of \(\mathrm{Out}(F_N)\), the concept of being pseudo-Anosov has several (nonequivalent) analogs. The first is the notion of an ‘atoroidal’ automorphism. An element \(\varphi\in\mathrm{Out}(F_N)\) is called atoroidal if there do not exist \(m\geq 1\), \(h\in F_N\), \(h\neq 1\) such that \(\varphi^m\) preserves the conjugacy class \([h]\) of \(h\) in \(F_N\). Another, more important, free group analog of being pseudo-Anosov is the notion of a ‘fully irreducible’ or ‘iwip’ automorphism. An element \(\varphi\in\mathrm{Out}(F_N)\) is called reducible if there exists a free product decomposition \(F_N=A_1*\cdots*A_k*C\) with \(k\geq 1\), \(A_i\neq 1\) and \(A_i\neq F_N\) such that \(\varphi\) permutes the conjugacy classes \([A_1],\ldots,[A_k]\). An element \(\varphi\in\mathrm{Out}(F_N)\) is irreducible if it is not reducible. An element \(\varphi\in\mathrm{Out}(F_N)\) is fully irreducible or iwip (which stands for ‘irreducible with irreducible powers’) if \(\varphi^m\) is irreducible for all integers \(m\geq 1\) (equivalently, for all nonzero integers \(m\)). Thus, \(\varphi\) is an iwip if and only if there do not exist a proper free factor \(A\) of \(F_N\) and \(m\geq 1\) such that \(\varphi^m([A])=[A]\). There is no obvious approach to algorithmically deciding whether an element\(\varphi\in\mathrm{Out}(F_N)\) is an iwip. In this note, we provide such an algorithm. Theorem A. There exists an algorithm that, given \(N\geq 2\) and \(\varphi\in\mathrm{Out}(F_N)\), decides whether or not \(\varphi\) is an iwip. Cited in 2 ReviewsCited in 15 Documents MSC: 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20E36 Automorphisms of infinite groups 20E05 Free nonabelian groups Keywords:algorithms; finitely generated free groups; iwip automorphisms; fully irreducible automorphisms PDFBibTeX XMLCite \textit{I. Kapovich}, Bull. Lond. Math. Soc. 46, No. 2, 279--290 (2014; Zbl 1319.20030) Full Text: DOI arXiv References: [1] Bestvina, A combination theorem for negatively curved groups, J. Differential Geom. 35 pp 85– (1992) · Zbl 0724.57029 [2] Bestvina, Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal. 7 pp 215– (1997) · Zbl 0884.57002 [3] Bestvina, The Tits alternative for Out(Fn). I. Dynamics of exponentially-growing automorphisms, Ann. of Math. 151 ((2)) pp 517– (2000) · Zbl 0984.20025 [4] Bestvina, The Tits alternative for Out(Fn). II. A Kolchin type theorem, Ann. of Math. 161 ((2)) pp 1– (2005) · Zbl 1139.20026 [5] Bestvina, Train tracks and automorphisms of free groups, Ann. of Math. 135 ((2)) pp 1– (1992) · Zbl 0757.57004 [6] Bestvina, Train-tracks for surface homeomorphisms, Topology 34 pp 109– (1995) · Zbl 0837.57010 [7] Bogopolski, Introduction to group theory (2008) · Zbl 1215.20001 [8] Bridson, The quadratic isoperimetric inequality for mapping tori of free group automorphisms, Mem. Amer. Math. Soc. 203 (2010) · Zbl 1201.20037 [9] Brinkmann, An implementation of the Bestvina-Handel algorithm for surface homeomorphisms, Experiment. Math. 9 pp 235– (2000) · Zbl 0982.57005 [10] Brinkmann, Hyperbolic automorphisms of free groups, Geom. Funct. Anal. 10 pp 1071– (2000) · Zbl 0970.20018 [11] Clay, Twisting out fully irreducible automorphisms, Geom. Funct. Anal. 20 pp 657– (2010) · Zbl 1206.20047 [12] Coulbois, Botany of irreducible automorphisms of free groups, Pacific J. Math. 256 pp 291– (2012) · Zbl 1259.20031 [13] Dicks, The group fixed by a family of injective endomorphisms of a free group (1996) · Zbl 0845.20018 [14] S. Dowdall I. Kapovich C. Leininger Dynamics on free-by-cyclic groups arXiv:1301.7739 [15] Gaboriau, An index for counting fixed points of automorphisms of free groups, Duke Math. J. 93 pp 425– (1998) · Zbl 0946.20010 [16] Guirardel, Dynamics of Out(Fn) on the boundary of outer space, Ann. Sci. École Norm. Sup. 33 ((4)) pp 433– (2000) · Zbl 1045.20034 [17] Jäger, Free group automorphisms with many fixed points at infinity, in: The Zieschang Gedenkschrift pp 321– (2008) [18] Kapovich, Ping-pong and Outer space, J. Topol. Anal. 2 pp 173– (2010) · Zbl 1211.20027 [19] Kapovich, Stallings foldings and the subgroup structure of free groups, J. Algebra 248 pp 608– (2002) · Zbl 1001.20015 [20] Levitt, Irreducible automorphisms of Fn have North-South dynamics on compactified outer space, J. Inst. Math. Jussieu 2 pp 59– (2003) · Zbl 1034.20038 [21] Papasoglu, An algorithm detecting hyperbolicity, in: Geometric and computational perspectives on infinite groups (Minneapolis, MN and New Brunswick, NJ, 1994) pp 193– (1996) [22] Pfaff, Ideal Whitehead graphs in Out(Fr) II: the complete graph in each rank, J. Homotopy Related Struct. · Zbl 1367.20049 [23] Stallings, Topology of finite graphs, Invent. Math. 71 pp 551– (1983) · Zbl 0521.20013 [24] Vogtmann, Automorphisms of free groups and outer space, Geom. Dedicata 94 pp 1– (2002) · Zbl 1017.20035 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.