Geodesic
Bootstrapping in convergence groups
Swenson, Eric L1
1Mathematics Department, Brigham Young University, Provo UT 84604, USA
Algebraic and Geometric Topology, Tome 5 (2005) no. 2, pp. 751-768
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

We prove a true bootstrapping result for convergence groups acting on a Peano continuum. We give an example of a Kleinian group H which is the amalgamation of two closed hyperbolic surface groups along a simple closed curve. The limit set ΛH is the closure of a “tree of circles" (adjacent circles meeting in pairs of points). We alter the action of H on its limit set such that H no longer acts as a convergence group, but the stabilizers of the circles remain unchanged, as does the action of a circle stabilizer on said circle. This is done by first separating the circles and then gluing them together backwards.

DOI : 10.2140/agt.2005.5.751
Keywords: convergence group, bootstrapping, Peano continuum

Swenson, Eric L  1

1 Mathematics Department, Brigham Young University, Provo UT 84604, USA 
@article{10_2140_agt_2005_5_751,
     author = {Swenson, Eric L},
     title = {Bootstrapping in convergence groups},
     journal = {Algebraic and Geometric Topology},
     pages = {751--768},
     year = {2005},
     volume = {5},
     number = {2},
     doi = {10.2140/agt.2005.5.751},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2005.5.751/}
}
TY  - JOUR
AU  - Swenson, Eric L
TI  - Bootstrapping in convergence groups
JO  - Algebraic and Geometric Topology
PY  - 2005
SP  - 751
EP  - 768
VL  - 5
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2005.5.751/
DO  - 10.2140/agt.2005.5.751
ID  - 10_2140_agt_2005_5_751
ER  - 
%0 Journal Article
%A Swenson, Eric L
%T Bootstrapping in convergence groups
%J Algebraic and Geometric Topology
%D 2005
%P 751-768
%V 5
%N 2
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2005.5.751/
%R 10.2140/agt.2005.5.751
%F 10_2140_agt_2005_5_751
Swenson, Eric L. Bootstrapping in convergence groups. Algebraic and Geometric Topology, Tome 5 (2005) no. 2, pp. 751-768. doi: 10.2140/agt.2005.5.751

[1] M Bestvina, G Mess, The boundary of negatively curved groups, J. Amer. Math. Soc. 4 (1991) 469

[2] B H Bowditch, A topological characterisation of hyperbolic groups, J. Amer. Math. Soc. 11 (1998) 643

[3] B H Bowditch, Peripheral splittings of groups, Trans. Amer. Math. Soc. 353 (2001) 4057

[4] B H Bowditch, Boundaries of geometrically finite groups, Math. Z. 230 (1999) 509

[5] B H Bowditch, Connectedness properties of limit sets, Trans. Amer. Math. Soc. 351 (1999) 3673

[6] J W Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984) 123

[7] J W Cannon, The theory of negatively curved spaces and groups, from: "Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989)", Oxford Sci. Publ., Oxford Univ. Press (1991) 315

[8] A Casson, D Jungreis, Convergence groups and Seifert fibered 3–manifolds, Invent. Math. 118 (1994) 441

[9] J Dugundji, Topology, Allyn and Bacon (1966)

[10] D Gabai, Convergence groups are Fuchsian groups, Ann. of Math. $(2)$ 136 (1992) 447

[11] M Gromov, Hyperbolic groups, from: "Essays in group theory", Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75

[12] E L Swenson, Axial pairs and convergence groups on $S^1$, Topology 39 (2000) 229

[13] E L Swenson, Convergence groups from subgroups, Geom. Topol. 6 (2002) 649

[14] P Tukia, Homeomorphic conjugates of Fuchsian groups, J. Reine Angew. Math. 391 (1988) 1

[15] P Tukia, Convergence groups and Gromov's metric hyperbolic spaces, New Zealand J. Math. 23 (1994) 157

[16] A Yaman, A topological characterisation of relatively hyperbolic groups, J. Reine Angew. Math. 566 (2004) 41

Cité par Sources :