Peripheral splittings of groups

Bowditch, B.

doi:10.1090/S0002-9947-01-02835-5

Peripheral splittings of groups
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by B. H. Bowditch

Trans. Amer. Math. Soc. 353 (2001), 4057-4082

DOI: https://doi.org/10.1090/S0002-9947-01-02835-5

Published electronically: June 1, 2001

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Abstract:

We define the notion of a “peripheral splitting” of a group. This is essentially a representation of the group as the fundamental group of a bipartite graph of groups, where all the vertex groups of one colour are held fixed—the “peripheral subgroups”. We develop the theory of such splittings and prove an accessibility result. The theory mainly applies to relatively hyperbolic groups with connected boundary, where the peripheral subgroups are precisely the maximal parabolic subgroups. We show that if such a group admits a non-trivial peripheral splitting, then its boundary has a global cut point. Moreover, the non-peripheral vertex groups of such a splitting are themselves relatively hyperbolic. These results, together with results from elsewhere, show that under modest constraints on the peripheral subgroups, the boundary of a relatively hyperbolic group is locally connected if it is connected. In retrospect, one further deduces that the set of global cut points in such a boundary has a simplicial treelike structure.

References

S. A. Adeleke and Peter M. Neumann, Relations related to betweenness: their structure and automorphisms, Mem. Amer. Math. Soc. 131 (1998), no. 623, viii+125. MR 1388893, DOI 10.1090/memo/0623
Mladen Bestvina and Mark Feighn, Bounding the complexity of simplicial group actions on trees, Invent. Math. 103 (1991), no. 3, 449–469. MR 1091614, DOI 10.1007/BF01239522
Mladen Bestvina and Geoffrey Mess, The boundary of negatively curved groups, J. Amer. Math. Soc. 4 (1991), no. 3, 469–481. MR 1096169, DOI 10.1090/S0894-0347-1991-1096169-1
B. H. Bowditch, Geometrical finiteness with variable negative curvature, Duke Math. J. 77 (1995), no. 1, 229–274. MR 1317633, DOI 10.1215/S0012-7094-95-07709-6
Jian-yi Shi, Left cells in the affine Weyl group of type $\widetilde F_4$, J. Algebra 200 (1998), no. 1, 173–206. MR 1603270, DOI 10.1006/jabr.1997.7043
B. H. Bowditch, Treelike structures arising from continua and convergence groups, Mem. Amer. Math. Soc. 139 (1999), no. 662, viii+86. MR 1483830, DOI 10.1090/memo/0662
B. H. Bowditch, Connectedness properties of limit sets, Trans. Amer. Math. Soc. 351 (1999), no. 9, 3673–3686. MR 1624089, DOI 10.1090/S0002-9947-99-02388-0
B.H. Bowditch, Relatively hyperbolic groups, preprint, Southampton (1997).
B. H. Bowditch, Boundaries of geometrically finite groups, Math. Z. 230 (1999), no. 3, 509–527. MR 1680044, DOI 10.1007/PL00004703
B.H. Bowditch, G.A. Swarup, Cut points in the boundaries of hyperbolic groups, in preparation.
Warren Dicks and M. J. Dunwoody, Groups acting on graphs, Cambridge Studies in Advanced Mathematics, vol. 17, Cambridge University Press, Cambridge, 1989. MR 1001965
M. J. Dunwoody, Cutting up graphs, Combinatorica 2 (1982), no. 1, 15–23. MR 671142, DOI 10.1007/BF02579278
M. J. Dunwoody, The accessibility of finitely presented groups, Invent. Math. 81 (1985), no. 3, 449–457. MR 807066, DOI 10.1007/BF01388581
M. J. Dunwoody, Groups acting on protrees, J. London Math. Soc. (2) 56 (1997), no. 1, 125–136. MR 1462830, DOI 10.1112/S0024610797005322
M. J. Dunwoody, Folding sequences, The Epstein birthday schrift, Geom. Topol. Monogr., vol. 1, Geom. Topol. Publ., Coventry, 1998, pp. 139–158. MR 1668347, DOI 10.2140/gtm.1998.1.139
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}
D.P. Guralnik, Constructing a splitting-tree for a cusp-finite group acting on a Peano continuum (Hebrew) : M.Sc. Dissertation, Technion, Haifa (1998).
André Haefliger, Complexes of groups and orbihedra, Group theory from a geometrical viewpoint (Trieste, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 504–540. MR 1170375
John L. Kelley, General topology, Graduate Texts in Mathematics, No. 27, Springer-Verlag, New York-Berlin, 1975. Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.]. MR 0370454
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
G. A. Swarup, On the cut point conjecture, Electron. Res. Announc. Amer. Math. Soc. 2 (1996), no. 2, 98–100. MR 1412948, DOI 10.1090/S1079-6762-96-00013-3
E.L. Swenson, A cutpoint tree for a continuum, in “Computational and Geometric Aspects of Modern Algebra”, London Math. Soc. Lecture Note Series, No. 275, (M. Atkinson, N. Gilbert, J. Howie, S. Linton, E. Robertson, eds.), Cambridge University Press (2000), 254–265.
Pekka Tukia, Conical limit points and uniform convergence groups, J. Reine Angew. Math. 501 (1998), 71–98. MR 1637829, DOI 10.1515/crll.1998.081
L. E. Ward Jr., Axioms for cutpoints, General topology and modern analysis (Proc. Conf., Univ. California, Riverside, Calif., 1980) Academic Press, New York-London, 1981, pp. 327–336. MR 619058