Abstract
We show first that certain automorphism groups of algebraic varieties, and even schemes, are residually finite and virtually torsion free. (A group virtually has a property if some subgroup of finite index has it.) The rest of the paper is devoted to a study of the groups of automorphisms. Aut(Γ) and outer automorphisms Out(Γ) of a finitely generated group Γ, by using the finite-dimensional representations of Γ. This is an old idea (cf. the discussion of Magnus in [11]). In particular the classes of semi-simplen-dimensional representations of Γ are parametrized by an algebraic varietyS n (Γ) on which Out(Γ) acts. We can apply the above results to this action and sometimes conclude that Out(Γ) is residually finite and virtually torsion free. This is true, for example, when Γ is a free group, or a surface group. In the latter case Out(Γ) is a “mapping class group.”
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. S. Birman,The algebraic structure of mapping class groups, inDiscrete Groups and Automorphic Functions (W. J. Harvey, ed.), Academic Press, 1977, pp. 163–198.
R. Fricke and F. Klein,Vorlesungen uber die Theorie der automorphen Functionen, Vol. 1, Leipzig, Teubner, 1897, pp. 365–370.
E. Grossman,On the residual finiteness of certain mapping class groups, J. London Math. Soc.9 (1974), 160–164.
A. Grothendieck (avec la collaboration de J. Dieudonné),Éléments de géométrie algébrique IV (Troisième partie), Publ. IHES28 (1966).
R. Horowitz,Characters of free groups represented in the two dimensional linear group, Comm. Pure Appl. Math.25 (1972), 635–649.
R. Horowitz,Induced automorphisms on Fricke characters of free groups, Trans. Am. Math. Soc.208 (1975), 41–50.
M. Jarden and J. Ritter,Normal automorphisms of absolute Galois groups of p-adic fields, Duke Math. J.47 (1980), 47–56.
A. Lubotzky,Nornial automorphisms of free groups, J. Algebra63 (1980), 494–498.
R. C. Lyndon and P. E. Schupp,Combinatorial Group Theory, Ergebnisse der Math. 89, Springer-Verlag, 1977.
A. M. Macbeath and D. Singerman,Spaces of subgroups and Teichmuller space, Proc. London Math. Soc.31 (1975), 211–256.
W. Magnus,Rings of Fricke characters and automorphism groups of free groups, Math. Z.170 (1980), 91–103.
J. Milnor,Introduction to Algebraic K-theory, Ann. Math. Studies, Princeton, 1971.
D. Mumford and K. Suominen,Introduction to the theory of moduli, inAlgebraic Geometry, Oslo, 1970 (F. Oort, ed.), Wolters-Noordhoff, Groningen, The Netherlands.
C. Procesi,Finite dimensional representations of algebras, Isr. J. Math.19 (1974), 169–182.
C. Procesi,Invariant theory of n by n matrices, Adv. Math.19 (1976), 306–381.
J. Smith,On products of profinite groups, Ill. J. Math.13 (1969), 680–688.
P. F. Stebe,Conjugacy separability of certain Fuchsian groups, Trans. Am. Math. Soc.163 (1972), 173–188.
H. Vogt,Sur les invariants fondamentaux des équations différentielles linéaires du seconde ordre, {jtAnn. Sci. École Norm. Sup.} {vn6} ({dy1889}), {snSuppl. 3–72}.
A. Whittemore,On special linear characters of free groups of rank n≧4, Proc. Am. Math. Soc.40 (1973), 383–388.
Author information
Authors and Affiliations
Additional information
Partially supported by the NSF under Grant MCS 80-05802.
Rights and permissions
About this article
Cite this article
Bass, H., Lubotzky, A. Automorphisms of groups and of schemes of finite type. Israel J. Math. 44, 1–22 (1983). https://doi.org/10.1007/BF02763168
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02763168