Available in electronic format
Available in print format		 Transacrions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

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

Author(s): Craig Jensen; Jon McCammond; John Meier
Journal: Trans. Amer. Math. Soc. 359 (2007), 2577-2595.
MSC (2000): Primary 20J06, 57M07
Posted: January 4, 2007
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

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:

[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)

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 20J06, 57M07

Retrieve articles in all Journals with MSC (2000): 20J06, 57M07

Additional Information:

Craig Jensen
Affiliation: Department of Mathematics, University of New Orleans, New Orleans, Louisiana 70148
Email: jensen@math.uno.edu

Jon McCammond
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
Email: jon.mccammond@math.ucsb.edu

John Meier
Affiliation: Department of Mathematics, Lafayette College, Easton, Pennsylvania 18042
Email: meierj@lafayette.edu

DOI: 10.1090/S0002-9947-07-03967-0
PII: S 0002-9947(07)03967-0
Received by editor(s): September 15, 2004
Received by editor(s) in revised form: February 9, 2005
Posted: 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
Copyright of article: Copyright 2007, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google