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ISSN: 1088-6834(e) ISSN: 0894-0347(p)
     

Splittings of Surfaces

Author(s): Richard K. Skora
Journal: J. Amer. Math. Soc. 9 (1996), 605-616.
MSC (1991): Primary 05C38, 15A15; Secondary 05A15, 15A18
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Abstract | References | Similar articles | Additional information

Abstract: The main result characterizes small actions of surface groups on ${\hbox{\normalshape{\bf R}}}$-trees.


References:

[Al-Ba]
R. Alperin and H. Bass, ``Length Functions of Group Actions on $\Lambda$-Trees," in: S. M. Gersten et al., eds., Combinatorial Group Theory and Topology, Annals of Math Studies #111, Princeton University Press, Princeton, (1987) 265-378. MR 89c:20057

[Ba]
H. Bass, ``Group Actions on Non-Archimedean Trees," in: R. C. Alperin, ed., Arboreal Group Theory, Mathematical Sciences Research Institute Publications, # 19, Springer-Verlag, New York (1991) 69-131. MR 93d:57003

[Cu-Mo]
M. Culler and J. W. Morgan, ``Group Actions on ${\hbox{\normalshape{\bf R}}}$-Trees," Proc. Lond. Math. Soc., 55, (1987) 571-604. MR 88f:20055

[FLP]
A. Fathi, F. Laudenbach, V. Poenaru, ``Travaux de Thurston sur les surfaces," Séminaire Orsay, Astérisque, 66-67 (1979) Soc. Math. de France, Paris. MR 82m:57003

[Gi-Sh]
H. Gillet and P. B. Shalen, ``Dendrology of Groups in low $\hbox{\normalshape{\bf Q}}$-ranks," J. Diff. Geo., 32 (1990) 605 - 712. MR 92b:57003

[Gi-Sh-Sk]
H. Gillet, P. B. Shalen and R. K. Skora, ``Simplicial Approximation and Low-rank Trees," Comment. Math. Helvetici, 66 (1991) 521-540. MR 93a:20038

[Ha]
A. Hatcher, ``Measured Lamination Spaces for Surfaces from the Topological Viewpoint," Top. Appl., 30, (1988), 63-88. MR 89k:57022

[Ji1]
R. Jiang, ``Branch Points and Free Actions on ${\hbox{\normalshape{\bf R}}}$-Trees," in: R. C. Alperin, ed., Arboreal Group Theory, Mathematical Sciences Research Institute Publications, # 19, Springer-Verlag, New York, (1991) 251-293. MR 92b:57001

[Ji2]
R. Jiang, ``On Free Actions on ${\hbox{\normalshape{\bf R}}}$-Trees," Ph.D. Thesis, Columbia University, (1990).

[L-S]
R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Bd. 89, Springer-Verlag, (1977). MR 58:28182

[Mo1]
J. W. Morgan, ``Group Actions on Trees and the Compactification of the Spaces of Classes of $SO(n,1)$-representations," Top., 25 (1986) 1-34. MR 87h:20062

[Mo2]
J. W. Morgan, ``Ergodic Theory and Free Actions on Trees," Invent. Math. 94 (1988) 605-622. MR 90d:20069

[Mo-Ot]
J. W. Morgan and J-P. Otal, ``Relative Growth Rates of Closed Geodesics on a Surface under varying Hyperbolic Structures," (preprint).

[Mo-Sh1]
J. W. Morgan and P. B. Shalen, ``Degenerations of Hyperbolic Structures, I: Valuations, Trees and Surfaces," Ann. Math., 120 (1984) 401-476. MR 86f:57011

[Mo-Sh2]
J. W. Morgan and P. B. Shalen, ``Degenerations of Hyperbolic Structures, II: Measured Laminations in $3$-Manifolds," Ann. of Math., 127 (1988) 403-456. MR 89e:57010a

[Mo-Sh3]
J. W. Morgan and P. B. Shalen, ``Degenerations of Hyperbolic Structures, III: Actions of $3$-Manifold Groups on Trees and Thurston's Compactness Theorem," Ann. of Math., 127 (1988) 457-519. MR 89e:57010b

[Mo-Sh4]
J. W. Morgan and P. B. Shalen, ``Free Actions of Surface Groups on ${\hbox{\normalshape{\bf R}}}$-Trees," Topology, 30 (1991) 143 - 154. MR 92e:20016

[Mo-Sk]
J. W. Morgan and R. K. Skora, ``Groups Acting Freely on $\hbox{\normalshape{\bf R}}$-trees," Ergodic Theory and Dynamical Systems, 11 (1991) 737-756. MR 94a:20041

[Pe-Ha]
R. Penner, with J. Harer, Combinatorics of Train Tracks, Annals of Mathematics Studies, # 125, Princeton University Press, Princeton (1992). MR 94b:57018

[Pla]
J. Plante, ``Foliations with Measure preserving Holonomy," Ann. of Math. 102 (1975) 327-361. MR 52:11947

[Ser]
J.-P. Serre, Trees, Springer-Verlag, New York (1980). MR 82c:20083

[Sh]
P. B. Shalen, ``Dendrology of Groups: An Introduction," in: S. M. Gersten ed., Essays in Group Theory, Mathematical Sciences Research Institute Publications, #8, Springer-Verlag, New York (1987) 265-319. MR 89d:57012

[Sk1]
R. K. Skora, ``Research Announcement : Splittings of Surfaces," Bull. A.M.S., 23 (1990) 85-90. MR 90k:57016

[Sk2]
R. K. Skora, ``Geometric Actions of Surface Groups on $\Lambda$-Trees," Comment. Math. Helvetici, 65 (1990) 517-533. MR 91k:20035

[Sk3]
R. K. Skora, ``Combination Theorems for Actions on Trees," (preprint, 1989).

[Sk4]
R. K. Skora, ``Deformations of Length Functions in Groups," (preprint, 1990).

[Th1]
W. Thurston, ``The Geometry and Topology of three-Manifolds," (preprint, 1979).

[Th2]
W. Thurston, ``Hyperbolic Structures on $3$-Manifolds, I: Deformation of acylindrical manifolds," Ann. Math., 124 (1986) 203-246. MR 88g:57014

[Th3]
W. Thurston, ``Hyperbolic Structures on $3$-Manifolds, II: Surface Groups and $3$-Manifolds which Fiber over the Circle," (preprint, 1986).


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Additional Information:

Richard K. Skora
Affiliation: Department of Mathematics, State University of New York, Stony Brook, New York 11794-3651 - Department of Mathematics, Columbia University, New York, New York 10027-6918
Email: rskora@interserv.com

DOI: 10.1090/S0894-0347-96-00205-6
PII: S 0894-0347(96)00205-6
Keywords: $\mathbf R$-tree, surface, hyperbolic space, splitting
Additional Notes: Supported in part by a SUNY Research Development Grant and an NSF Postdoctoral Research Fellowship.
Copyright of article: Copyright 1996, American Mathematical Society


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