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The Euler characteristic of the Whitehead automorphism group of a free product

Authors: Craig Jensen, Jon McCammond and John Meier
Journal: Trans. Amer. Math. Soc. 359 (2007), 2577-2595
MSC (2000): Primary 20J06, 57M07
Published electronically: January 4, 2007
MathSciNet review: 2286046
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Abstract: A combinatorial summation identity over the lattice of labelled hypertrees is established that allows one to gain concrete information on the Euler characteristics of various automorphism groups of free products of groups. In particular, we establish formulae for the Euler characteristics of: the group of Whitehead automorphisms $ \mathrm{Wh}(\ast_{i=1}^n G_i)$ when the $ G_i$ are of finite homological type; $ \operatorname{Aut}(\ast_{i=1}^n G_i)$ and $ \operatorname{Out} (\ast_{i=1}^n G_i)$ when the $ G_i$ are finite; and the palindromic automorphism groups of finite rank free groups.

References [Enhancements On Off] (What's this?)

  • [Br74] Kenneth S. Brown.
    Euler characteristics of discrete groups and $ G$-spaces.
    Invent. Math., 27:229-264, 1974. MR 0385007 (52:5877)
  • [Br94] Kenneth S. Brown.
    Cohomology of groups, volume 87 of Graduate Texts in Mathematics.
    Springer-Verlag, New York, 1994.
    Corrected reprint of the 1982 original. MR 1324339 (96a:20072)
  • [Co89] Donald J. Collins.
    Cohomological dimension and symmetric automorphisms of a free group.
    Comment. Math. Helv., 64(1):44-61, 1989. MR 0982561 (90e:20035)
  • [Co95] Donald J. Collins.
    Palindromic automorphisms of free groups.
    In Combinatorial and geometric group theory (Edinburgh, 1993), volume 204 of London Math. Soc. Lecture Note Ser., pages 63-72. Cambridge Univ. Press, Cambridge, 1995. MR 1320275 (96c:20048)
  • [CL83] Donald J. Collins and Frank Levin.
    Automorphisms and Hopficity of certain Baumslag-Solitar groups.
    Arch. Math. (Basel), 40(5):385-400, 1983. MR 0707725 (85b:20043)
  • [CV86] Marc Culler and Karen Vogtmann.
    Moduli of graphs and automorphisms of free groups.
    Invent. Math., 84(1):91-119, 1986. MR 0830040 (87f:20048)
  • [GJ00] Henry H. Glover and Craig A. Jensen.
    Geometry for palindromic automorphism groups of free groups.
    Comment. Math. Helv., 75(4):644-667, 2000. MR 1789180 (2002m:20058)
  • [GKP94] Ronald L. Graham, Donald E. Knuth, and Oren Patashnik.
    Concrete mathematics.
    Addison-Wesley Publishing Company, Reading, MA, second edition, 1994.
    A foundation for computer science. MR 1397498 (97d:68003)
  • [Ha71] G. Harder.
    A Gauss-Bonnet formula for discrete arithmetically defined groups.
    Ann. Sci. École Norm. Sup. (4), 4:409-455, 1971. MR 0309145 (46:8255)
  • [HZ86] J. Harer and D. Zagier.
    The Euler characteristic of the moduli space of curves.
    Invent. Math., 85(3):457-485, 1986. MR 0848681 (87i:32031)
  • [KV93] Sava Krstic and Karen Vogtmann.
    Equivariant outer space and automorphisms of free-by-finite groups.
    Comment. Math. Helv., 68(2):216-262, 1993. MR 1214230 (94c:20067)
  • [MM04] Jon McCammond and John Meier.
    The hypertree poset and the $ l\sp 2$-Betti numbers of the motion group of the trivial link.
    Math. Ann., 328(4):633-652, 2004. MR 2047644 (2005b:20104)
  • [MM96] Darryl McCullough and Andy Miller.
    Symmetric automorphisms of free products.
    Mem. Amer. Math. Soc., 122(582):viii+97, 1996. MR 1329943 (96k:20069)
  • [Pe88] R. C. Penner.
    Perturbative series and the moduli space of Riemann surfaces.
    J. Differential Geom., 27(1):35-53, 1988. MR 0918455 (89h:32045)
  • [Ro84] Steven Roman.
    The umbral calculus, volume 111 of Pure and Applied Mathematics.
    Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1984. MR 0741185 (87c:05015)
  • [Se79] J.-P. Serre.
    Arithmetic groups.
    In Homological group theory (Proc. Sympos., Durham, 1977), volume 36 of London Math. Soc. Lecture Note Ser., pages 105-136. Cambridge Univ. Press, Cambridge, 1979. MR 0564421 (82b:22021)
  • [SV87] John Smillie and Karen Vogtmann.
    A generating function for the Euler characteristic of $ {\rm Out}(F\sb n)$.
    J. Pure Appl. Algebra, 44(1-3):329-348, 1987. MR 0885116 (88g:20107)
  • [St99] Richard P. Stanley.
    Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics.
    Cambridge University Press, Cambridge, 1999.
    With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282 (2000k:05026)

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Additional Information

Craig Jensen
Affiliation: Department of Mathematics, University of New Orleans, New Orleans, Louisiana 70148

Jon McCammond
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106

John Meier
Affiliation: Department of Mathematics, Lafayette College, Easton, Pennsylvania 18042

Received by editor(s): September 15, 2004
Received by editor(s) in revised form: February 9, 2005
Published electronically: January 4, 2007
Additional Notes: The first author was partially supported by Louisiana Board of Regents RCS contract no. LEQSF-RD-A-39
The second author was partially supported by NSF grant no. DMS-0101506
The third author was partially supported by an AMS Centennial Research Fellowship
Article copyright: © Copyright 2007 American Mathematical Society

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