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On subgroups of finite complexity in groups acting on trees. (English) Zbl 1107.20021
Suppose that \(G\) is a group acting on a tree \(X\). Given a subgroup \(H\) of \(G\), the author studies the existence of invariants of \(H\), uniquely determined by the action of \(H\) on \(X\). For this he describes some properties of subgroups of finite complexity, an extension of the Schreier index formula for the rank of subgroups of finite index in free groups, to the case of subgroups of finite index in amalgamated free products. He also shows that all isomorphic subgroups of finite index in a finitely presented group with infinitely many ends have the same index, and that fixed subgroups of symmetric endomorphisms of fundamental groups of graphs of groups whose edge groups are all finite, are tame. Finally, the chains of subgroups of finite complexity are described.

MSC:
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20E07 Subgroup theorems; subgroup growth
20E08 Groups acting on trees
20F05 Generators, relations, and presentations of groups
20F65 Geometric group theory
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[1] Bass, H., Some remarks on group actions on trees, Comm. algebra, 4, 1091-1126, (1976) · Zbl 0383.20021
[2] Bass, H., Covering theory for graphs of groups, J. pure appl. algebra, 89, 3-47, (1993) · Zbl 0805.57001
[3] K.S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer, Berlin, 1982. · Zbl 0584.20036
[4] Burns, R.G., On the finitely generated subgroups of an amalgamated product of two groups, Trans. amer. math. soc., 169, 293-306, (1972) · Zbl 0254.20020
[5] Burns, R.G., Finitely generated subgroups of HNN groups, Canad. J. math., 25, 1103-1112, (1973) · Zbl 0238.20033
[6] Burns, R.G.; Chau, T.C.; Kam, S.-M., On the rank of intersection of subgroups of a free product of groups, J. pure appl. algebra, 124, 31-45, (1998) · Zbl 0896.20021
[7] Cohen, D.E., Subgroups of HNN groups, J. austral. math. soc., 17, 394-405, (1974) · Zbl 0288.20041
[8] Collins, D.J.; Turner, E.C., Efficient representatives for automorphisms of free products, Michigan math. J., 41, 443-464, (1994) · Zbl 0820.20035
[9] Dicks, W.; Dunwoody, M.J., Group acting on graphs, (1989), Cambridge University Press Cambridge · Zbl 0665.20001
[10] Dunwoody, M.J., The accessibility of finitely presented groups, Invent. math., 81, 449-457, (1985) · Zbl 0572.20025
[11] Greenberg, L., Commensurable groups of moebius transformations, Ann. math. studies, 79, 227-237, (1974) · Zbl 0295.20054
[12] Griffiths, H.B., The fundamental group of a surface, and a theorem of Schreier, Acta math., 110, 1-17, (1963) · Zbl 0119.18902
[13] Karrass, A.; Pietrowski, A.; Solitar, D., Automorphisms of a free product with an amalgamated subgroup, contributions to group theory, Contemp. math., 33, 328-340, (1984)
[14] Karrass, A.; Solitar, D., The subgroups of a free product of two groups with an amalgamated subgroup, Trans. amer. math. soc., 150, 227-255, (1970) · Zbl 0223.20031
[15] Karrass, A.; Solitar, D., Subgroups of HNN groups and groups with one defining relation, Canad. J. math., 23, 627-643, (1971) · Zbl 0232.20051
[16] Kulkarni, R., An extension of a theorem of kurosh and applications to Fuchsian groups, Michigan math. J., 30, 259-272, (1983) · Zbl 0571.20039
[17] Mihalik, M.; Towle, W., Quasiconvex subgroups of negatively curved groups, J. pure appl. algebra, 95, 297-301, (1994) · Zbl 0822.20038
[18] J. Nielsen, The commutator group of the free product of cyclic groups, Mat. Tidsskr. B. (1948) 49-56.
[19] Reznikov, A., Volumes of discrete groups and topological complexity of homology spheres, Math. ann., 306, 547-554, (1996) · Zbl 0859.20027
[20] Scott, P., Finitely generated 3-manifold groups are finitely presented, J. London math. soc., 6, 2, 437-440, (1973) · Zbl 0254.57003
[21] P. Scott, G.A. Swarup, Regular neighbourhoods and canonical decompositions for groups, Asterisque No. 289 (2003) vi+233 pp. · Zbl 1036.20028
[22] P. Scott, C.T.C. Wall, Topological methods in group theory, in: Homological Group Theory, London Mathematical Society Lecture Notes Series, vol. 36, 1979, pp. 137-214. · Zbl 0423.20023
[23] Serre, J.P., Trees, (1980), Springer New York
[24] Stallings, J., Topology of finite graphs, Invent. math., 71, 551-565, (1983) · Zbl 0521.20013
[25] Sykiotis, M., Fixed points of symmetric endomorphisms of groups, Internat. J. algebra comput., 12, 5, 737-745, (2002) · Zbl 1010.20016
[26] Sykiotis, M., Stable representatives for symmetric automorphisms of groups and the general form of the Scott conjecture, Trans. amer. math. soc., 356, 2405-2441, (2004) · Zbl 1041.20019
[27] M. Sykiotis, Fixed subgroups of endomorphisms of free products, in preparation. · Zbl 1130.20032
[28] M. Sykiotis, The complexity volume of splittable groups, in preparation. · Zbl 1107.20021
[29] Takahasi, M., Note on chain conditions in free groups, Osaka math. J., 3, 221-225, (1951) · Zbl 0044.01106
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