Algebraic & Geometric Topology

Automorphisms of free groups with boundaries

Craig Jensen and Nathalie Wahl

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The automorphisms of free groups with boundaries form a family of groups An,k closely related to mapping class groups, with the standard automorphisms of free groups as An,0 and (essentially) the symmetric automorphisms of free groups as A0,k. We construct a contractible space Ln,k on which An,k acts with finite stabilizers and finite quotient space and deduce a range for the virtual cohomological dimension of An,k. We also give a presentation of the groups and calculate their first homology group.

Article information

Algebr. Geom. Topol., Volume 4, Number 1 (2004), 543-569.

Received: 26 February 2004
Revised: 8 July 2004
Accepted: 9 July 2004
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20F28: Automorphism groups of groups [See also 20E36]
Secondary: 20F05: Generators, relations, and presentations

automorphism groups classifying spaces.


Jensen, Craig; Wahl, Nathalie. Automorphisms of free groups with boundaries. Algebr. Geom. Topol. 4 (2004), no. 1, 543--569. doi:10.2140/agt.2004.4.543.

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  • N. Brady, J. McCammond, J. Meier, and A. Miller, The pure symmetric automorphisms of a free group form a duality group, J. Algebra 246 (2001) 881-896.
  • K. Brown, Cohomology of Groups, Springer-Verlag, New York-Berlin, 1982.
  • A. Brownstein and R. Lee, Cohomology of the group of motions of $n$ strings in 3-space, Contemp. Math. vol. 150 (1993) 51-61.
  • D. J. Collins, Cohomological dimension and symmetric automorphisms of a free group, Comment. Math. Helv. 64 (1989) 44-61.
  • M. Culler and K. Vogtmann, Moduli spaces of graphs and automorphisms of free groups, Invent. Math. 84 (1986) 91-119.
  • D. Dahm, A generalization of braid theory, Ph.D. Thesis, Princeton Univ., 1962.
  • D. Goldsmith, The theory of motion groups, Mich. Math. J. 28 (1981), 3-17.
  • J. L. Harer, Stability of the homology of the mapping class groups of orientable surfaces, Annals Math. 121 (1985), 215-249.
  • A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.
  • A. Hatcher and K. Vogtmann, Cerf theory for graphs, J. London Math. Soc. (2) 58 (1998) 633-655.
  • A. Hatcher and N. Wahl, Stabilization for the automorphisms of free groups with boundary, preprint..
  • C. A. Jensen, Cohomology of $\Aut(F_n)$ in the $p$-rank two case, J. Pure Appl. Algebra 158 (2001) 41-81.
  • C. A. Jensen, Contractibility of fixed point sets of auter space, Topology Appl. 119 (2002) 287-304.
  • C. A. Jensen, J. McCammond and J. Meier, The Euler characteristic of the Whitehead automorphism group of a free product, preprint.
  • C. A. Jensen, Homology of holomorphs of free groups, J. Algebra 271 (2004) 281-294.
  • S. Krstić, Actions of finite groups on graphs and related automorphisms of free groups, J. Algebra 124 (1989) 119-138.
  • S. Krstić and K. Vogtmann, Equivariant outer space and automorphisms of free-by-finite groups, Comment. Math. Helv. 68 (1993) 216-262.
  • G. Levitt, Automorphisms of hyperbolic groups and graphs of groups, preprint..
  • I. Madsen and U. Tillmann, The stable mapping class group and $\mathcal{Q}(\mathbb{C}P^\infty_+)$, Invent. math. 145 (2001), 509-544.
  • I. Madsen and M. Weiss, The stable moduli space of Riemann surfaces: Mumford's conjecture, preprint..
  • W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory. Presentations of groups in terms of generators and relations, Second edition, Dover Pub., New York, 1976.
  • J. McCammond and J. Meier, The hypertree poset and the $L^2$ Betti numbers of the motion group of the trivial link, Math. Ann. 328 (2004) 633-652.
  • J. McCool, A presentation for the automorphism group of a free group of finite rank, J. London Math. Soc. (2) 8 (1974),259-266.
  • J. McCool, Some finitely presented subgroups of the automorphism group of a free group, J. Algebra 35 (1975), 205-213.
  • J. McCool, On basis-conjugating automorphisms of the free groups, Can. J. Math., 38 (1986) no. 6, 1525-1529.
  • D. McCullough and A. Miller, Symmetric Automorphisms of Free Products, Mem. Amer. Math. Soc. 122 (1996), no. 582.
  • D. Quillen, Homotopy properties of the poset of non-trivial $p$-subgroups of a group, Advances in Math. 28 (1978) 101-128.
  • U. Tillmann, On the homotopy of the stable mapping class group, Invent. Math. 130 (1997), 257-275.
  • N. Wahl, From mapping class groups to automorphisms of free groups, preprint..