Automorphism groups of polycyclic-by-finite groups and arithmetic groups
Publications Mathématiques de l'IHÉS, Volume 104 (2006), pp. 213-268.

We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construct examples of polycyclic-by-finite groups which have an automorphism group which does not contain an arithmetic group of finite index. Finally we discuss applications of our results to the groups of homotopy self-equivalences of K(Γ,1)-spaces and obtain an extension of arithmeticity results of Sullivan in rational homotopy theory.

DOI: 10.1007/s10240-006-0003-3
Baues, Oliver 1; Grunewald, Fritz 

1 ETH-Zürich, Departement Mathematik, Rämistrasse 101, Ch-8092 Zürich
@article{PMIHES_2006__104__213_0,
     author = {Baues, Oliver and Grunewald, Fritz},
     title = {Automorphism groups of polycyclic-by-finite groups and arithmetic groups},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {213--268},
     publisher = {Springer},
     volume = {104},
     year = {2006},
     doi = {10.1007/s10240-006-0003-3},
     zbl = {1121.20027},
     mrnumber = {2264837},
     language = {en},
     url = {http://www.numdam.org/articles/10.1007/s10240-006-0003-3/}
}
TY  - JOUR
AU  - Baues, Oliver
AU  - Grunewald, Fritz
TI  - Automorphism groups of polycyclic-by-finite groups and arithmetic groups
JO  - Publications Mathématiques de l'IHÉS
PY  - 2006
DA  - 2006///
SP  - 213
EP  - 268
VL  - 104
PB  - Springer
UR  - http://www.numdam.org/articles/10.1007/s10240-006-0003-3/
UR  - https://zbmath.org/?q=an%3A1121.20027
UR  - https://www.ams.org/mathscinet-getitem?mr=2264837
UR  - https://doi.org/10.1007/s10240-006-0003-3
DO  - 10.1007/s10240-006-0003-3
LA  - en
ID  - PMIHES_2006__104__213_0
ER  - 
%0 Journal Article
%A Baues, Oliver
%A Grunewald, Fritz
%T Automorphism groups of polycyclic-by-finite groups and arithmetic groups
%J Publications Mathématiques de l'IHÉS
%D 2006
%P 213-268
%V 104
%I Springer
%U https://doi.org/10.1007/s10240-006-0003-3
%R 10.1007/s10240-006-0003-3
%G en
%F PMIHES_2006__104__213_0
Baues, Oliver; Grunewald, Fritz. Automorphism groups of polycyclic-by-finite groups and arithmetic groups. Publications Mathématiques de l'IHÉS, Volume 104 (2006), pp. 213-268. doi : 10.1007/s10240-006-0003-3. http://www.numdam.org/articles/10.1007/s10240-006-0003-3/

1. L. Auslander, The automorphism group of a polycyclic group, Ann. Math. (2), 89 (1969), 314-322 | MR | Zbl

2. L. Auslander, On a problem of Philip Hall, Ann. Math. (2), 86 (1967), 112-116 | MR | Zbl

3. L. Auslander, F.E.A. Johnson, On a conjecture of C. T. C. Wall, J. Lond. Math. Soc., II. Ser., 14 (1976), 331-332 | MR | Zbl

4. L. Auslander, G. Baumslag, Automorphism groups of finitely generated nilpotent groups, Bull. Am. Math. Soc., 73 (1967), 716-717 | MR | Zbl

5. G. Baumslag, Automorphism groups of nilpotent groups, Am. J. Math., 91 (1969), 1003-1011 | MR | Zbl

6. G. Baumslag, Lectures on Nilpotent Groups, A.M.S., Providence, R.I. (1971)

7. O. Baues, Finite extensions and unipotent shadows of affine crystallographic groups, C. R. Acad. Sci., Paris, Sér. I, Math., 335 (2002), 785-788 | MR | Zbl

8. O. Baues, Infrasolvmanifolds and rigidity of subgroups in solvable linear algebraic groups, Topology, 43 (2004), 903-924 | MR | Zbl

9. A. Borel, Arithmetic properties of linear algebraic groups, Proc. I.C.M. Stockholm (1962), 10-22. | MR | Zbl

10. A. Borel, Density and maximality of arithmetic subgroups, J. Reine Angew. Math., 224 (1966), 78-89 | MR | Zbl

11. A. Borel, Linear algebraic groups, second edn., Graduate Texts in Mathematics, vol. 126, Springer, New York, 1991. | MR | Zbl

12. A. Borel, J.-P. Serre, Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv., 39 (1964), 111-164 | MR | Zbl

13. A. Borel, J. Tits, Groupes réductifs, Publ. Math., Inst. Hautes Étud. Sci., 27 (1965), 55-150 | Numdam | MR | Zbl

14. K.S. Brown, Cohomology of groups, Springer, New York-Berlin (1982) | MR | Zbl

15. R.M. Bryant, J.R.J. Groves, Algebraic groups of automorphisms of nilpotent groups and Lie algebras, J. Lond. Math. Soc., II. Ser., 33 (1986), 453-466 | MR | Zbl

16. P. Deligne, Extensions centrales non résiduellement finies de groupes arithmétiques, C. R. Acad. Sci., Paris, Sér. A-B, 287 (1978), A203-A208 | MR | Zbl

17. M. Du Sautoy, Polycyclic groups, analytic groups and algebraic groups, Proc. Lond. Math. Soc., III. Ser., 85 (2002), 62-92 | MR | Zbl

18. P.A. Griffiths, J.W. Morgan, Rational homotopy theory and differential forms, Birkhäuser, Boston, Mass. (1981) | MR | Zbl

19. F. Grunewald, V. Platonov, Solvable arithmetic groups and arithmeticity problems, Duke Math. J., 10 (1999), 327-366 | MR | Zbl

20. F. Grunewald, V. Platonov, On finite extensions of arithmetic groups, C. R. Acad. Sci. Paris, 325 (1997), 1153-1158 | MR | Zbl

21. F. Grunewald, V. Platonov, Rigidity results for groups with radical, cohomology of finite groups and arithmeticity problems, Duke Math. J., 100 (1999), 321-358 | MR | Zbl

22. F. Grunewald, V. Platonov, Non-arithmetic polycyclic groups, C. R. Acad. Sci. Paris, 326 (1998), 1359-1364 | MR | Zbl

23. F. Grunewald, V. Platonov, Rigidity and automorphism groups of solvable arithmetic groups, C. R. Acad. Sci. Paris, 327 (1998), 427-432 | MR | Zbl

24. F. Grunewald, J. O'Halloran, Nilpotent groups and unipotent algebraic groups, J. Pure Appl. Algebra, 37 (1985), 299-313 | Zbl

25. F. Grunewald, D. Segal, On affine crystallographic groups, J. Differ. Geom., 40 (1994), 563-594 | MR | Zbl

26. J.-L. Koszul, Homologie et cohomologie des algébres de Lie, Bull. Soc. Math. Fr., 78 (1950), 65-127 | Numdam | MR | Zbl

27. L.A. Lambe, S.B. Priddy, Cohomology of nilmanifolds and torsion-free, nilpotent groups, Trans. Am. Math. Soc., 273 (1982), 39-55 | MR | Zbl

28. S. Maclane, Homology, Springer, Berlin-Göttingen-Heidelberg (1963) | MR | Zbl

29. A. I. Mal'cev, On a class of homogeneous spaces, Am. Math. Soc. Transl., 39 (1951), 1-33.

30. I. Ju. Merzljakov, Integer representation of the holomorphs of polycyclic groups, Algebra Log., 9 (1970), 539-558 | MR | Zbl

31. G.D. Mostow, Representative functions on discrete groups and solvable arithmetic subgroups, Am. J. Math., 92 (1970), 1-32 | MR | Zbl

32. G.D. Mostow, Some applications of representative functions to solvmanifolds, Am. J. Math., 93 (1971), 11-32 | MR | Zbl

33. K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. Math. (2), 59 (1954), 531-538 | MR | Zbl

34. P.F. Pickel, J. Roitberg, Automorphism groups of nilpotent groups and spaces, J. Pure Appl. Algebra, 150 (2000), 307-319 | MR | Zbl

35. V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Academic Press, Boston, MA, 1994. | MR | Zbl

36. M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 68, Springer, New York-Heidelberg, 1972. | MR | Zbl

37. J. Roitberg, Genus and symmetry in homotopy theory, Math. Ann., 305 (1996), 381-386 | MR | Zbl

38. V.A. Romankov, Width of verbal subgroups in solvable groups, Algebra Log., 21 (1982), 60-72 | MR | Zbl

39. J. W. Rutter, Spaces of homotopy self-equivalences, Lect. Notes Math., vol. 1662, Springer, Berlin, 1997. | MR | Zbl

40. D. Segal, Polycyclic groups, Cambridge Univ. Press, London (1983) | MR | Zbl

41. D. Segal, On the outer automorphism group of a polycyclic group, Proceedings of the Second International Group Theory Conference (Bressanone, 1989), Rend. Circ. Mat. Palermo, II. Ser. (1990), Suppl. no. 23, 265-278. | MR | Zbl

42. J.-P. Serre, Arithmetic groups, Lond. Math. Soc. Lect. Notes, 36 (1979), 105-135 | MR | Zbl

43. J.-P. Serre, Cohomologie des groupes discrets, Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970), pp. 77-169, Ann. Math. Stud., no. 70, Princeton Univ. Press, Princeton, N.J., 1971. | MR | Zbl

44. D. Sullivan, Infinitesimal computations in topology, Publ. Math., Inst. Hautes Étud. Sci., 47 (1977), 269-331 | Numdam | MR | Zbl

45. D. Sullivan, Genetics of homotopy theory and the Adams conjecture, Ann. Math. (2), 100 (1974), 1-79 | MR | Zbl

46. B.A.F. Wehrfritz, Two remarks on polycyclic groups, Bull. Lond. Math. Soc., 26 (1994), 543-548 | MR | Zbl

47. B.A.F. Wehrfritz, On the holomorphs of soluble groups of finite rank, J. Pure Appl. Algebra, 4 (1974), 55-69 | MR | Zbl

Cited by Sources: