# zbMATH — the first resource for mathematics

Peripheral fillings of relatively hyperbolic groups. (English) Zbl 1116.20031
The author first notes that Thurston’s Dehn filling theorem for 3-manifolds implies the following group theoretic result: If $$G$$ is the fundamental group of a complete finite volume hyperbolic 3-manifold and if $$H_1,\dots,H_k$$ are the cusp subgroups of $$G$$, then there exists a finite subset $$\mathcal F$$ of $$G$$ such that for any collection of primitive elements $$x_i$$ in $$H_i\setminus\mathcal F$$, the quotient group $$G/\langle x_1,\dots,x_k\rangle^G$$ is word hyperbolic. (Here, if $$S$$ is a subset of $$G$$, $$\langle S\rangle^G$$ denotes the normal closure of $$S$$ in $$G$$.)
In the paper under review, the author gives the following generalized setting of this result. Instead of the fundamental group of a complete finite volume hyperbolic manifold, he considers a general relatively hyperbolic group $$G$$. In this setting, the algebraic analogue of Dehn filling is defined as follows. Let $$\{H_\lambda\}_{\lambda\in\Lambda}$$ be a collection of subgroups of $$G$$. To each collection of groups $$\mathcal N=\{N_\lambda\}_{\lambda\in\Lambda}$$, where for each $$\lambda$$, $$N_\lambda$$ is a normal subgroup of $$H_\lambda$$, one associates the quotient group $G(\mathcal N)=G/\langle\bigcup_{\lambda\in\Lambda}N_\lambda\rangle^G.$ Furthermore, instead of taking a single element $$x_i$$ in $$H_i$$, the author considers normal subgroups generated by arbitrary subsets of the cusp subgroups.
The main result of this paper is then the following Theorem: Let $$G$$ be a group which is relatively hyperbolic with respect to a collection of subgroups $$\{H_\lambda\}_{\lambda\in\Lambda}$$. Then there exists a finite subset $$\mathcal F$$ of non-trivial elements of $$G$$ with the following property. Let $$\mathcal N=\{N_\lambda\}_{\lambda\in\Lambda}$$ be a collection of subgroups, where each $$N_\lambda$$ is normal in $$H_\lambda$$ and such that $$N_\lambda\cap\mathcal F=\emptyset$$ for all $$\lambda$$ in $$\Lambda$$. Then, for each $$\lambda$$ in $$\Lambda$$, the natural map $$H_\lambda/N_\lambda\to G(\mathcal N)$$ is injective, and the quotient group $$G(\mathcal N)$$ is hyperbolic relative to the collection $$\{H_\lambda/N_\lambda\}_{\lambda\in\Lambda}$$. Moreover, for any finite subset $$S\subset G$$, there exists a finite subset $$\mathcal F(S)$$ of non-trivial elements of $$G$$ such that the restriction of the natural homomorphism $$G\to G(\mathcal N)$$ to $$S$$ is injective whenever $$N_\lambda\cap\mathcal F(S)=\emptyset$$ for all $$\lambda\in\Lambda$$.
Note that in this theorem, $$G$$ is not assumed to be finitely generated.
The proof of the theorem is combinatorial and uses techniques related to van Kampen diagrams.
The author deduces several interesting corollaries, among them the following Corollary: Suppose, in addition to the assumptions of the theorem, that $$G$$ is finitely generated and $$H_\lambda/N_\lambda$$ is hyperbolic for each $$\lambda\in\Lambda$$. Then, $$G(\mathcal N)$$ is hyperbolic.
The author also obtains algebraic applications of the theorem. For instance, he proves that any almost malnormal quasi-convex subgroup of a hyperbolic group “almost has the congruence extension property”. This is related to some work of Brian Bowditch. He also obtains non-trivial examples of fully residually hyperbolic groups, and he proves that the following two assertions are equivalent: (1) All hyperbolic groups are residually finite. (2) If a finitely generated group $$G$$ is hyperbolic relative to a collection of residually finite subgroups, then $$G$$ is residually finite.
Note that the question of whether all hyperbolic groups are residually finite is an open question.

##### MSC:
 20F65 Geometric group theory 20F67 Hyperbolic groups and nonpositively curved groups 20F06 Cancellation theory of groups; application of van Kampen diagrams 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 20E26 Residual properties and generalizations; residually finite groups
Full Text:
##### References:
 [1] Alonso, J., Bridson, M.: Semihyperbolic groups. Proc. Lond. Math. Soc., III. Ser. 70, 56–114 (1995) · Zbl 0823.20035 [2] Anderson, M.T.: Dehn filling and Einstein metrics in higher dimensions. J. Differ. Geom. 73, 219–261 (2006) · Zbl 1100.53039 [3] Blok, W.J., Pigozzi, D.: On the congruence extension property. Algebra Univers. 38, 391–394 (1997) · Zbl 0934.08003 [4] Bowditch, B.H.: Relatively hyperbolic groups. Prep. 1999 · Zbl 0952.20032 [5] Bridson, M., Haefliger, A.: Metric Spaces of Non-Positive Curvature. Berlin: Springer 1999 · Zbl 0988.53001 [6] Druţu, C., Sapir, M.: Tree-graded spaces and asymptotic cones of groups. With an appendix by D. Osin and M. Sapir. Topology 44, 959–1058 (2005) · Zbl 1101.20025 [7] Eberlein, P.: Lattices in spaces of nonpositive curvature. Ann. Math. 111, 435–476 (1980) · Zbl 0432.53023 [8] Farb, B.: Relatively hyperbolic groups. Geom. Funct. Anal. 8, 810–840 (1998) · Zbl 0985.20027 [9] Gromov, M.: Hyperbolic groups, Essays in Group Theory. MSRI Series, S.M. Gersten, ed., vol. 8, pp. 75–263. New York: Springer 1987 [10] Groves, D.: Limits of (certain) CAT(0) groups, II: The Hopf property and the shortening argument. Prep. 2004; available at arXiv: math.GR/0408080 [11] Groves, D.: Limits of (certain) CAT(0) groups, I: Compactification. Prep. 2004; available at arXiv: math.GR/0404440. · Zbl 1085.20025 [12] Groves, D., Manning, J.: Dehn filling in relatively hyperbolic groups. Prep. 2006; available at arXiv: math.GR/0601311 · Zbl 1211.20038 [13] Groves, D., Manning, J.: Fillings, finite generation, and direct limits of relatively hyperbolic groups. Prep. 2006; available at arXiv: math.GR/0606070 · Zbl 1169.20023 [14] Guirardel, V.: Limit groups and groups acting freely on $$\mathbb{R}$$ n -trees. Geom. Topol. 8, 1427–1470 (2004) · Zbl 1114.20013 [15] Hall, P.: The Edmonton notes on nilpotent groups. Queen Mary College Mathematics Notes. London: Mathematics Department, Queen Mary College 1969 · Zbl 0211.34201 [16] Higman, G., Neumann, B.H., Neumann, H.: Embedding theorems for groups. J. Lond. Math. Soc. 24, 247–254 (1949) · Zbl 0034.30101 [17] Ivanov, S.V., Olshanskii, A.Y.: Hyperbolic groups and their quotients of bounded exponents. Trans. Am. Math. Soc. 348, 2091–2138 (1996) · Zbl 0876.20023 [18] Karrass, A., Magnus, W., Solitar, D.: Elements of finite order in groups with a single defining relation. Commun. Pure Appl. Math. 13, 57–66 (1960) · Zbl 0091.02403 [19] Kharlampovich, O., Myasnikov, A.: Description of fully residually free groups and irreducible affine varieties over a free group. Summer School in Group Theory in Banff, 1996, CRM Proc. Lecture Notes, vol. 17, pp. 71–80. Providence, RI: Amer. Math. Soc. 1999 · Zbl 0922.20027 [20] Lyndon, R.C., Shupp, P.E.: Combinatorial Group Theory. Berlin–New York: Springer 1977 [21] Olshanskii, A.Y.: Geometry of defining relations in groups. Mathematics and its Applications (Soviet Series), vol. 70. Dordrecht: Kluwer Academic Publishers Group 1991 [22] Olshanskii, A.Y.: Periodic quotients of hyperbolic groups. Math. USSR Sbornik 72, 519–541 (1992) · Zbl 0820.20044 [23] Olshanski, A.Y.: SQ-universality of hyperbolic groups, (Russian). Mat. Sb. 186, 119–132 (1995) [24] Olshanskii, A.Y., Sapir, M.V.: Non-amenable finitely presented torsion-by-cyclic groups. Publ. Math., Inst. Hautes Étud. Sci. 96, 43–169 (2002) · Zbl 1050.20019 [25] Osin, D.V.: Relatively Hyperbolic Groups: Intrinsic Geometry, Algebraic Properties, and Algorithmic Problems. Mem. Am. Math. Soc. vol. 179. Providence, RI: American Mathematical Society 2006 · Zbl 1093.20025 [26] Osin, D.V.: Elementary subgroups of hyperbolic groups and bounded generation. Int. J. Algebra Comput. 16, 99–118 (2006) · Zbl 1100.20033 [27] Osin, D.V.: Asymptotic dimension of relatively hyperbolic groups. Int. Math. Res. Not. 2005, 2143–2162 (2005) · Zbl 1089.20028 [28] Ozawa, N.: Boundary amenability of relatively hyperbolic groups. Prep. 2005; available at http://www.arxiv.org/abs/math.GR/0501555 · Zbl 1109.20037 [29] Rebbechi, Y.D.: Algorithmic Properties of Relatively Hyperbolic Groups. PhD thesis, Rutgers Uniw. (Newark); available at http://www.arxiv.org/abs/math.GR/0302245 [30] Rolfsen, D.: Knots and Links. Math. Lect. Series, vol. 7. Houston, TX: Publish or Perish Inc. 1976 · Zbl 0339.55004 [31] Sela, Z.: Diophantine geometry over groups I: Makanin–Razborov diagrams. Publ. Math., Inst. Hautes Étud. Sci. 93, 31–105 (2001) · Zbl 1018.20034 [32] Stallings, J.: Group theory and three-dimensional manifolds. Yale Mathematical Monographs, vol. 4. New Haven, Conn.-London: Yale University Press 1971 · Zbl 0241.57001 [33] Tang, X.: Semigroups with the congruence extension property. Semigroup Forum 56, 228–264 (1998) · Zbl 0981.20050 [34] Thurston, W.P.: Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Am. Math. Soc., New Ser. 6, 357–381 (1982) · Zbl 0496.57005 [35] Tukia, P.: Convergence groups and Gromov’s metric hyperbolic spaces. N. Z. J. Math. 23, 157–187 (1994) · Zbl 0855.30036 [36] Wise, D.: The residual finiteness of negatively curved polygons of finite groups. Invent. Math. 149, 579–617 (2002) · Zbl 1040.20024 [37] Wu, M.: The fuzzy congruence extension property in groups. JP J. Algebra Number Theory Appl. 2, 153–160 (2002) · Zbl 1032.20045 [38] Yaman, A.: A topological characterization of relatively hyperbolic groups. J. Reine Angew. Math. 566, 41–89 (2004) · Zbl 1043.20020 [39] Yu, G.: The Novikov conjecture for groups with finite asymptotic dimension. Ann. Math. (2) 147 325–355 (1998) · Zbl 0911.19001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.