On a theorem of Scott and Swarup
HTML articles powered by AMS MathViewer
- by Mahan Mitra PDF
- Proc. Amer. Math. Soc. 127 (1999), 1625-1631 Request permission
Abstract:
Let $1 \rightarrow H \rightarrow G \rightarrow \mathbb {Z} \rightarrow 1$ be an exact sequence of hyperbolic groups induced by an automorphism $\phi$ of the free group $H$. Let $H_1 ( \subset H)$ be a finitely generated distorted subgroup of $G$. Then there exist $N > 0$ and a free factor $K$ of $H$ such that the conjugacy class of $K$ is preserved by $\phi ^N$ and $H_1$ contains a finite index subgroup of a conjugate of $K$. This is an analog of a theorem of Scott and Swarup for surfaces in hyperbolic 3-manifolds.References
- M. Bestvina and M. Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992), no. 1, 85–101. MR 1152226, DOI 10.4310/jdg/1214447806
- M. Bestvina, M. Feighn, and M. Handel. The Tits’ alternative for Out($F_n$) I: Dynamics of exponentially growing automorphisms. preprint.
- M. Bestvina, M. Feighn, and M. Handel, Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal. 7 (1997), no. 2, 215–244. MR 1445386, DOI 10.1007/PL00001618
- Mladen Bestvina and Michael Handel, Train tracks and automorphisms of free groups, Ann. of Math. (2) 135 (1992), no. 1, 1–51. MR 1147956, DOI 10.2307/2946562
- J. Cannon and W. P. Thurston. Group Invariant Peano Curves. preprint.
- Benson Farb, The extrinsic geometry of subgroups and the generalized word problem, Proc. London Math. Soc. (3) 68 (1994), no. 3, 577–593. MR 1262309, DOI 10.1112/plms/s3-68.3.577
- M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR 1253544
- M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
- M. Mitra. Ending Laminations for Hyperbolic Group Extensions. Geom. Funct. Anal. vol.7 No. 2, pages 379–402, 1997.
- M. Mitra. PhD Thesis, U.C.Berkeley. 1997.
- M. Mitra. Cannon-Thurston Maps for Hyperbolic Group Extensions. Topology, 1998.
- Peter Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2) 17 (1978), no. 3, 555–565. MR 494062, DOI 10.1112/jlms/s2-17.3.555
- G. P. Scott and G. A. Swarup, Geometric finiteness of certain Kleinian groups, Proc. Amer. Math. Soc. 109 (1990), no. 3, 765–768. MR 1013981, DOI 10.1090/S0002-9939-1990-1013981-6
- Hamish Short, Quasiconvexity and a theorem of Howson’s, Group theory from a geometrical viewpoint (Trieste, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 168–176. MR 1170365
Additional Information
- Mahan Mitra
- Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720
- Address at time of publication: Institute of Mathematical Sciences, C.I.T. Campus, Madras (Chennai) - 600113, India
- Email: mitra@imsc.ernet.in
- Received by editor(s): September 22, 1997
- Published electronically: February 17, 1999
- Additional Notes: The author’s research was partly supported by an Alfred P. Sloan Doctoral Dissertation Fellowship, Grant No. DD 595
- Communicated by: Christopher Croke
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 1625-1631
- MSC (1991): Primary 20F32, 57M50
- DOI: https://doi.org/10.1090/S0002-9939-99-04935-7
- MathSciNet review: 1610757