On suspensions and conjugacy of hyperbolic automorphisms

Dahmani, François

doi:10.1090/tran/6530

On suspensions and conjugacy of hyperbolic automorphisms
HTML articles powered by AMS MathViewer

by François Dahmani PDF

Trans. Amer. Math. Soc. 368 (2016), 5565-5577 Request permission

Abstract:

We remark that the conjugacy problem for pairs of hyperbolic automorphisms of a finitely presented group (typically a free group) is decidable. The solution that we propose uses the isomorphism problem for the suspensions, and the study of their automorphism group.

References

Goulnara Arzhantseva, Jean-François Lafont, and Ashot Minasyan, Isomorphism versus commensurability for a class of finitely presented groups, J. Group Theory 17 (2014), no. 2, 361–378. MR 3176665, DOI 10.1515/jgt-2013-0050
O. Bogopolski, A. Martino, and E. Ventura, Orbit decidability and the conjugacy problem for some extensions of groups, Trans. Amer. Math. Soc. 362 (2010), no. 4, 2003–2036. MR 2574885, DOI 10.1090/S0002-9947-09-04817-X
Hyman Bass, Covering theory for graphs of groups, J. Pure Appl. Algebra 89 (1993), no. 1-2, 3–47. MR 1239551, DOI 10.1016/0022-4049(93)90085-8
P. Brinkmann, Hyperbolic automorphisms of free groups, Geom. Funct. Anal. 10 (2000), no. 5, 1071–1089. MR 1800064, DOI 10.1007/PL00001647
Peter Brinkmann, Splittings of mapping tori of free group automorphisms, Geom. Dedicata 93 (2002), 191–203. MR 1934698, DOI 10.1023/A:1020329530768
François Dahmani, Existential questions in (relatively) hyperbolic groups, Israel J. Math. 173 (2009), 91–124. MR 2570661, DOI 10.1007/s11856-009-0084-z
François Dahmani and Vincent Guirardel, Foliations for solving equations in groups: free, virtually free, and hyperbolic groups, J. Topol. 3 (2010), no. 2, 343–404. MR 2651364, DOI 10.1112/jtopol/jtq010
François Dahmani and Vincent Guirardel, The isomorphism problem for all hyperbolic groups, Geom. Funct. Anal. 21 (2011), no. 2, 223–300. MR 2795509, DOI 10.1007/s00039-011-0120-0
François Dahmani and Daniel Groves, The isomorphism problem for toral relatively hyperbolic groups, Publ. Math. Inst. Hautes Études Sci. 107 (2008), 211–290. MR 2434694, DOI 10.1007/s10240-008-0014-3
Jérôme E. Los, On the conjugacy problem for automorphisms of free groups, Topology 35 (1996), no. 3, 779–808. With an addendum by the author. MR 1396778, DOI 10.1016/0040-9383(95)00035-6
M. Lustig, Structure and conjugacy for automorphisms of free groups I, MPI-Bonn preprint series 2000, No. 241; http://www.mpim-bonn.mpg.de/preprints
M. Lustig, Structure and conjugacy for automorphisms of free groups II, MPI-Bonn preprint series 2001, No. 4; http://www.mpim-bonn.mpg.de/preprints
Martin Lustig, Conjugacy and centralizers for iwip automorphisms of free groups, Geometric group theory, Trends Math., Birkhäuser, Basel, 2007, pp. 197–224. MR 2395795, DOI 10.1007/978-3-7643-8412-8_{1}1
P. Papasoglu, An algorithm detecting hyperbolicity, Geometric and computational perspectives on infinite groups (Minneapolis, MN and New Brunswick, NJ, 1994) DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 25, Amer. Math. Soc., Providence, RI, 1996, pp. 193–200. MR 1364185
Z. Sela, The isomorphism problem for hyperbolic groups. I, Ann. of Math. (2) 141 (1995), no. 2, 217–283. MR 1324134, DOI 10.2307/2118520
Jean-Pierre Serre, Arbres, amalgames, $\textrm {SL}_{2}$, Astérisque, No. 46, Société Mathématique de France, Paris, 1977 (French). Avec un sommaire anglais; Rédigé avec la collaboration de Hyman Bass. MR 0476875
John Stallings, Group theory and three-dimensional manifolds, Yale Mathematical Monographs, vol. 4, Yale University Press, New Haven, Conn.-London, 1971. A James K. Whittemore Lecture in Mathematics given at Yale University, 1969. MR 0415622