Geometry & Topology

Residual properties of automorphism groups of (relatively) hyperbolic groups

Gilbert Levitt and Ashot Minasyan

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We show that Out(G) is residually finite if G is one-ended and hyperbolic relative to virtually polycyclic subgroups. More generally, if G is one-ended and hyperbolic relative to proper residually finite subgroups, the group of outer automorphisms preserving the peripheral structure is residually finite. We also show that Out(G) is virtually residually p–finite for every prime p if G is one-ended and toral relatively hyperbolic, or infinitely-ended and virtually residually p–finite.

Article information

Geom. Topol., Volume 18, Number 5 (2014), 2985-3023.

Received: 10 June 2013
Revised: 5 February 2014
Accepted: 8 March 2014
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 20F28: Automorphism groups of groups [See also 20E36] 20E26: Residual properties and generalizations; residually finite groups

relatively hyperbolic groups outer automorphism groups residually finite


Levitt, Gilbert; Minasyan, Ashot. Residual properties of automorphism groups of (relatively) hyperbolic groups. Geom. Topol. 18 (2014), no. 5, 2985--3023. doi:10.2140/gt.2014.18.2985.

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  • E Alibegović, A combination theorem for relatively hyperbolic groups, Bull. London Math. Soc. 37 (2005) 459–466
  • R B J T Allenby, G Kim, C Y Tang, Residual finiteness of outer automorphism groups of certain pinched $1$–relator groups, J. Algebra 246 (2001) 849–858
  • M Aschenbrenner, S Friedl, $3$–manifold groups are virtually residually $p$, Mem. Amer. Math. Soc. 1058, Amer. Math. Soc. (2013)
  • B Baumslag, Residually free groups, Proc. London Math. Soc. 17 (1967) 402–418
  • G Baumslag, On generalised free products, Math. Z. 78 (1962) 423–438
  • G Baumslag, Automorphism groups of residually finite groups, J. London Math. Soc. 38 (1963) 117–118
  • B H Bowditch, Relatively hyperbolic groups, Internat. J. Algebra Comput. 22 (2012) 1250016, 66
  • I Bumagin, D T Wise, Every group is an outer automorphism group of a finitely generated group, J. Pure Appl. Algebra 200 (2005) 137–147
  • D E Cohen, Combinatorial group theory: A topological approach, London Math. Soc. Student Texts 14, Cambridge Univ. Press (1989)
  • F Dahmani, Combination of convergence groups, Geom. Topol. 7 (2003) 933–963
  • W Dicks, M J Dunwoody, Groups acting on graphs, Cambridge Studies in Advanced Mathematics 17, Cambridge Univ. Press (1989)
  • J L Dyer, Separating conjugates in free-by-finite groups, J. London Math. Soc. 20 (1979) 215–221
  • J L Dyer, Separating conjugates in amalgamated free products and HNN extensions, J. Austral. Math. Soc. Ser. A 29 (1980) 35–51
  • L Funar, Two questions on mapping class groups, Proc. Amer. Math. Soc. 139 (2011) 375–382
  • E K Grossman, On the residual finiteness of certain mapping class groups, J. London Math. Soc. 9 (1974/75) 160–164
  • D Groves, J F Manning, Dehn filling in relatively hyperbolic groups, Israel J. Math. 168 (2008) 317–429
  • V Guirardel, G Levitt, JSJ decompositions: definitions, existence, uniqueness, II: Compatibility and acylindricity
  • V Guirardel, G Levitt, McCool groups of toral relatively hyperbolic groups, In preparation
  • V Guirardel, G Levitt, Splittings and automorphisms of relatively hyperbolic groups, to appear in Groups, Geometry, Dynamics
  • V Guirardel, G Levitt, The outer space of a free product, Proc. Lond. Math. Soc. 94 (2007) 695–714
  • B Hartley, On residually finite $p$–groups, from: “Symposia Mathematica, Vol. XVII”, Academic Press, London (1976) 225–234
  • G C Hruska, Relative hyperbolicity and relative quasiconvexity for countable groups, Algebr. Geom. Topol. 10 (2010) 1807–1856
  • G Levitt, Automorphisms of hyperbolic groups and graphs of groups, Geom. Dedicata 114 (2005) 49–70
  • A Lubotzky, Normal automorphisms of free groups, J. Algebra 63 (1980) 494–498
  • A Martino, A proof that all Seifert $3$–manifold groups and all virtual surface groups are conjugacy separable, J. Algebra 313 (2007) 773–781
  • V Metaftsis, M Sykiotis, On the residual finiteness of outer automorphisms of relatively hyperbolic groups, J. Pure Appl. Algebra 214 (2010) 1301–1305
  • A Minasyan, D Osin, Normal automorphisms of relatively hyperbolic groups, Trans. Amer. Math. Soc. 362 (2010) 6079–6103
  • A Minasyan, D Osin, Fixed subgroups of automorphisms of relatively hyperbolic groups, Q. J. Math. 63 (2012) 695–712
  • D V Osin, Relative Dehn functions of amalgamated products and HNN–extensions, from: “Topological and asymptotic aspects of group theory”, (R Grigorchuk, M Mihalik, M Sapir, Z Šuni\'k, editors), Contemp. Math. 394, Amer. Math. Soc. (2006) 209–220
  • D V Osin, Relatively hyperbolic groups: Intrinsic geometry, algebraic properties and algorithmic problems, Mem. Amer. Math. Soc. 179 (2006)
  • D V Osin, Peripheral fillings of relatively hyperbolic groups, Invent. Math. 167 (2007) 295–326
  • L Paris, Residual $p$ properties of mapping class groups and surface groups, Trans. Amer. Math. Soc. 361 (2009) 2487–2507
  • V N Remeslennikov, Finite approximability of groups with respect to conjugacy, Sibirsk. Mat. Ž. 12 (1971) 1085–1099 In Russian; translated in Siberian Math. J. 23 (1971) 783–792
  • A H Rhemtulla, Residually $F\sb{p}$–groups, for many primes $p$, are orderable, Proc. Amer. Math. Soc. 41 (1973) 31–33
  • D J S Robinson, A course in the theory of groups, 2nd edition, Graduate Texts in Mathematics 80, Springer (1996)
  • P Scott, The geometries of $3$–manifolds, Bull. London Math. Soc. 15 (1983) 401–487
  • J-P Serre, Arbres, amalgames, $\mathrm{SL}_{2}$, Astérisque 46, Soc. Math. France, Paris (1977)
  • J Stallings, Group theory and three-dimensional manifolds, Yale Mathematical Monographs 4, Yale University Press (1971)
  • P F Stebe, Conjugacy separability of certain free products with amalgamation, Trans. Amer. Math. Soc. 156 (1971) 119–129
  • W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979) Available at \setbox0\makeatletter\@url {\unhbox0
  • E Toinet, Conjugacy $p$–separability of right-angled Artin groups and applications, Groups Geom. Dyn. 7 (2013) 751–790
  • B A F Wehrfritz, Infinite linear groups: An account of the group-theoretic properties of infinite groups of matrices, Ergebnisse der Matematik 76, Springer (1973)
  • B A F Wehrfritz, Two remarks on polycyclic groups, Bull. London Math. Soc. 26 (1994) 543–548