Automorphism groups of polycyclic-by-finite groups and arithmetic groups

Oliver Baues[1]; Fritz Grunewald

  • [1] ETH-Zürich, Departement Mathematik, Rämistrasse 101, Ch-8092 Zürich

Publications Mathématiques de l'IHÉS (2006)

  • Volume: 104, page 213-268
  • ISSN: 0073-8301

Abstract

top
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.

How to cite

top

Baues, Oliver, and Grunewald, Fritz. "Automorphism groups of polycyclic-by-finite groups and arithmetic groups." Publications Mathématiques de l'IHÉS 104 (2006): 213-268. <http://eudml.org/doc/104220>.

@article{Baues2006,
abstract = {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.},
affiliation = {ETH-Zürich, Departement Mathematik, Rämistrasse 101, Ch-8092 Zürich},
author = {Baues, Oliver, Grunewald, Fritz},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {polycyclic-by-finite groups; outer automorphism groups; arithmetic groups; rational homotopy theory},
language = {eng},
pages = {213-268},
publisher = {Springer},
title = {Automorphism groups of polycyclic-by-finite groups and arithmetic groups},
url = {http://eudml.org/doc/104220},
volume = {104},
year = {2006},
}

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
PB - Springer
VL - 104
SP - 213
EP - 268
AB - 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.
LA - eng
KW - polycyclic-by-finite groups; outer automorphism groups; arithmetic groups; rational homotopy theory
UR - http://eudml.org/doc/104220
ER -

References

top
  1. 1. L. Auslander, The automorphism group of a polycyclic group, Ann. Math. (2), 89 (1969), 314-322 Zbl0197.29902MR271202
  2. 2. L. Auslander, On a problem of Philip Hall, Ann. Math. (2), 86 (1967), 112-116 Zbl0149.26904MR218454
  3. 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 Zbl0364.22008MR423362
  4. 4. L. Auslander, G. Baumslag, Automorphism groups of finitely generated nilpotent groups, Bull. Am. Math. Soc., 73 (1967), 716-717 Zbl0149.26905MR217168
  5. 5. G. Baumslag, Automorphism groups of nilpotent groups, Am. J. Math., 91 (1969), 1003-1011 Zbl0208.03203MR255654
  6. 6. G. Baumslag, Lectures on Nilpotent Groups, A.M.S., Providence, R.I. (1971) 
  7. 7. O. Baues, Finite extensions and unipotent shadows of affine crystallographic groups, C. R. Acad. Sci., Paris, Sér. I, Math., 335 (2002), 785-788 Zbl1023.20025MR1947699
  8. 8. O. Baues, Infrasolvmanifolds and rigidity of subgroups in solvable linear algebraic groups, Topology, 43 (2004), 903-924 Zbl1059.57022MR2061212
  9. 9. A. Borel, Arithmetic properties of linear algebraic groups, Proc. I.C.M. Stockholm (1962), 10–22. Zbl0134.16502MR175901
  10. 10. A. Borel, Density and maximality of arithmetic subgroups, J. Reine Angew. Math., 224 (1966), 78-89 Zbl0158.03105MR205999
  11. 11. A. Borel, Linear algebraic groups, second edn., Graduate Texts in Mathematics, vol. 126, Springer, New York, 1991. Zbl0726.20030MR1102012
  12. 12. A. Borel, J.-P. Serre, Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv., 39 (1964), 111-164 Zbl0143.05901MR181643
  13. 13. A. Borel, J. Tits, Groupes réductifs, Publ. Math., Inst. Hautes Étud. Sci., 27 (1965), 55-150 Zbl0145.17402MR207712
  14. 14. K.S. Brown, Cohomology of groups, Springer, New York–Berlin (1982) Zbl0584.20036MR672956
  15. 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 Zbl0554.20008MR850961
  16. 16. P. Deligne, Extensions centrales non résiduellement finies de groupes arithmétiques, C. R. Acad. Sci., Paris, Sér. A-B, 287 (1978), A203-A208 Zbl0416.20042MR507760
  17. 17. M. du Sautoy, Polycyclic groups, analytic groups and algebraic groups, Proc. Lond. Math. Soc., III. Ser., 85 (2002), 62-92 Zbl1025.20022MR1901369
  18. 18. P.A. Griffiths, J.W. Morgan, Rational homotopy theory and differential forms, Birkhäuser, Boston, Mass. (1981) Zbl0474.55001MR641551
  19. 19. F. Grunewald, V. Platonov, Solvable arithmetic groups and arithmeticity problems, Duke Math. J., 10 (1999), 327-366 Zbl1039.20026MR1688145
  20. 20. F. Grunewald, V. Platonov, On finite extensions of arithmetic groups, C. R. Acad. Sci. Paris, 325 (1997), 1153-1158 Zbl0913.20034MR1490116
  21. 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 Zbl1007.11029MR1722957
  22. 22. F. Grunewald, V. Platonov, Non-arithmetic polycyclic groups, C. R. Acad. Sci. Paris, 326 (1998), 1359-1364 Zbl0974.20035MR1649174
  23. 23. F. Grunewald, V. Platonov, Rigidity and automorphism groups of solvable arithmetic groups, C. R. Acad. Sci. Paris, 327 (1998), 427-432 Zbl0914.20043MR1652546
  24. 24. F. Grunewald, J. O’Halloran, Nilpotent groups and unipotent algebraic groups, J. Pure Appl. Algebra, 37 (1985), 299-313 Zbl0576.20019
  25. 25. F. Grunewald, D. Segal, On affine crystallographic groups, J. Differ. Geom., 40 (1994), 563-594 Zbl0822.20050MR1305981
  26. 26. J.-L. Koszul, Homologie et cohomologie des algébres de Lie, Bull. Soc. Math. Fr., 78 (1950), 65-127 Zbl0039.02901MR36511
  27. 27. L.A. Lambe, S.B. Priddy, Cohomology of nilmanifolds and torsion-free, nilpotent groups, Trans. Am. Math. Soc., 273 (1982), 39-55 Zbl0507.22008MR664028
  28. 28. S. Maclane, Homology, Springer, Berlin–Göttingen–Heidelberg (1963) Zbl0133.26502MR156879
  29. 29. A. I. Mal’cev, On a class of homogeneous spaces, Am. Math. Soc. Transl., 39 (1951), 1–33. 
  30. 30. I. Ju. Merzljakov, Integer representation of the holomorphs of polycyclic groups, Algebra Log., 9 (1970), 539-558 Zbl0221.20043MR280578
  31. 31. G.D. Mostow, Representative functions on discrete groups and solvable arithmetic subgroups, Am. J. Math., 92 (1970), 1-32 Zbl0205.04406MR271267
  32. 32. G.D. Mostow, Some applications of representative functions to solvmanifolds, Am. J. Math., 93 (1971), 11-32 Zbl0228.22015MR283819
  33. 33. K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. Math. (2), 59 (1954), 531-538 Zbl0058.02202MR64057
  34. 34. P.F. Pickel, J. Roitberg, Automorphism groups of nilpotent groups and spaces, J. Pure Appl. Algebra, 150 (2000), 307-319 Zbl0962.20026MR1769359
  35. 35. V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Academic Press, Boston, MA, 1994. Zbl0841.20046MR1278263
  36. 36. M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 68, Springer, New York–Heidelberg, 1972. Zbl0254.22005MR507234
  37. 37. J. Roitberg, Genus and symmetry in homotopy theory, Math. Ann., 305 (1996), 381-386 Zbl0864.55004MR1391222
  38. 38. V.A. Romankov, Width of verbal subgroups in solvable groups, Algebra Log., 21 (1982), 60-72 Zbl0504.20020MR683939
  39. 39. J. W. Rutter, Spaces of homotopy self-equivalences, Lect. Notes Math., vol. 1662, Springer, Berlin, 1997. Zbl0889.55004MR1474967
  40. 40. D. Segal, Polycyclic groups, Cambridge Univ. Press, London (1983) Zbl0516.20001MR713786
  41. 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. Zbl0703.20034MR1068367
  42. 42. J.-P. Serre, Arithmetic groups, Lond. Math. Soc. Lect. Notes, 36 (1979), 105-135 Zbl0432.20042MR564421
  43. 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. Zbl0235.22020MR385006
  44. 44. D. Sullivan, Infinitesimal computations in topology, Publ. Math., Inst. Hautes Étud. Sci., 47 (1977), 269-331 Zbl0374.57002MR646078
  45. 45. D. Sullivan, Genetics of homotopy theory and the Adams conjecture, Ann. Math. (2), 100 (1974), 1-79 Zbl0355.57007MR442930
  46. 46. B.A.F. Wehrfritz, Two remarks on polycyclic groups, Bull. Lond. Math. Soc., 26 (1994), 543-548 Zbl0819.20036MR1315604
  47. 47. B.A.F. Wehrfritz, On the holomorphs of soluble groups of finite rank, J. Pure Appl. Algebra, 4 (1974), 55-69 Zbl0286.20031MR347985

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.