Inventiones mathematicae

, Volume 215, Issue 2, pp 713–740 | Cite as

Simple groups separated by finiteness properties

  • Rachel SkipperEmail author
  • Stefan Witzel
  • Matthew C. B. Zaremsky


We show that for every positive integer n there exists a simple group that is of type \(\mathrm {F}_{n-1}\) but not of type \(\mathrm {F}_n\). For \(n\ge 3\) these groups are the first known examples of this kind. They also provide infinitely many quasi-isometry classes of finitely presented simple groups. The only previously known infinite family of such classes, due to Caprace–Rémy, consists of non-affine Kac–Moody groups over finite fields. Our examples arise from Röver–Nekrashevych groups, and contain free abelian groups of infinite rank.

Mathematics Subject Classification

Primary 20E32 Secondary 57M07 20F65 20E08 



We are grateful to Dessislava Kochloukova and Said Sidki for sharing a preprint of [36] with us and to Bertrand Rémy for asking the question that motivated our search for simple groups separated by finiteness properties. We also thank Matt Brin, Pierre-Emmanuel Caprace, Eduard Schesler, and Marco Varisco for many helpful comments.


  1. 1.
    Abels, H., Brown, K.S.: Finiteness properties of solvable \(S\)-arithmetic groups: an example. In: Proceedings of the Northwestern Conference on Cohomology of Groups (Evanston, Illinois, 1985), vol. 44, pp. 77–83 (1987)Google Scholar
  2. 2.
    Alonso, J.M.: Finiteness conditions on groups and quasi-isometries. J. Pure Appl. Algebra 95(2), 121–129 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Abramenko, P., Mühlherr, B.: Présentations de certaines \(BN\)-paires jumelées comme sommes amalgamées. C. R. Acad. Sci. Paris Sér. I Math. 325(7), 701–706 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Artin, E.: Algebraic Numbers and Algebraic Functions. Gordon and Breach, New York (1967)zbMATHGoogle Scholar
  5. 5.
    Bestvina, M., Brady, N.: Morse theory and finiteness properties of groups. Invent. Math. 129(3), 445–470 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bell, G., Dranishnikov, A.: Asymptotic dimension. Topol. Appl. 155(12), 1265–1296 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Belk, J., Forrest, B.: Rearrangement groups of fractals. Trans. Am. Math. Soc. (to appear). arXiv:1510.03133v2
  8. 8.
    Bux, K.-U., Fluch, M.G., Marschler, M., Witzel, S., Zaremsky, M.C.B.: The braided Thompson’s groups are of type \(\text{ F }_\infty \). J. Reine Angew. Math. 718, 59–101 (2016). With an appendix by ZaremskyMathSciNetzbMATHGoogle Scholar
  9. 9.
    Brown, K.S., Geoghegan, R.: An infinite-dimensional torsion-free \(\text{ FP }_{\infty }\) group. Invent. Math. 77(2), 367–381 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bieri, R., Geoghegan, R., Kochloukova, D.H.: The sigma invariants of Thompson’s group \(F\). Groups Geom. Dyn. 4(2), 263–273 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bieri, R.: Homological Dimension of Discrete Groups. Mathematics Department, Queen Mary College, London (1976). Queen Mary College Mathematics NoteszbMATHGoogle Scholar
  12. 12.
    Bux, K.-U., Köhl, R., Witzel, S.: Higher finiteness properties of reductive arithmetic groups in positive characteristic: the rank theorem. Ann. Math. (2) 177(1), 311–366 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Burger, M., Mozes, S.: Lattices in product of trees. Inst. Hautes Études Sci. Publ. Math. 92, 151–194 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Belk, J., Matucci, F.: Röver’s simple group is of type \(F_\infty \). Publ. Math. 60(2), 501–524 (2016)CrossRefzbMATHGoogle Scholar
  15. 15.
    Bux, K.-U., Mohammadi, A., Wortman, K.: \(\text{ SL }_n({\mathbb{Z}}[t])\) is not \(\text{ FP }_{n-1}\). Comment. Math. Helv. 85(1), 151–164 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Bartholdi, L., Neuhauser, M., Woess, W.: Horocyclic products of trees. J. Eur. Math. Soc. 10(3), 771–816 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Brin, M.G.: Higher dimensional Thompson groups. Geom. Dedicata 108, 163–192 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Brin, M.G.: On the baker’s map and the simplicity of the higher dimensional Thompson groups \(nV\). Publ. Math. 54(2), 433–439 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Brown, K.S.: Finiteness properties of groups. In: Proceedings of the Northwestern Conference on Cohomology of Groups (Evanston, Illinois, 1985), vol. 44, pp. 45–75 (1987)Google Scholar
  20. 20.
    Bux, K.-U.: Finiteness properties of certain metabelian arithmetic groups in the function field case. Proc. Lond. Math. Soc. (3) 75(2), 308–322 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Bux, K.-U.: Finiteness properties of soluble arithmetic groups over global function fields. Geom. Topol. 8, 611–644 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Camm, R.: Simple free products. J. Lond. Math. Soc. 28, 66–76 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Cannon, J.W., Floyd, W.J., Parry, W.R.: Introductory notes on Richard Thompson’s groups. Enseign. Math. (2) 42(3–4), 215–256 (1996)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Caprace, P.-E., Rémy, B.: Simplicity and superrigidity of twin building lattices. Invent. Math. 176(1), 169–221 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Caprace, P.-E., Rémy, B.: Non-distortion of twin building lattices. Geom. Dedicata 147, 397–408 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Dymara, J., Schick, T.: Buildings have finite asymptotic dimension. Russ. J. Math. Phys. 16(3), 409–412 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Farley, D.S.: Finiteness and \(\text{ CAT }(0)\) properties of diagram groups. Topology 42(5), 1065–1082 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Fluch, M.G., Marschler, M., Witzel, S., Zaremsky, M.C.B.: The Brin–Thompson groups \(sV\) are of type \(\text{ F }_\infty \). Pac. J. Math. 266(2), 283–295 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Gandini, G.: Bounding the homological finiteness length. Bull. Lond. Math. Soc. 44(6), 1209–1214 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Grigorchuk, R.I.: On the system of defining relations and the Schur multiplier of periodic groups generated by finite automata. In: Groups St. Andrews 1997 in Bath, I, volume 260 of London Mathematical Society Lecture Note Series, pp. 290–317. Cambridge University Press, Cambridge (1999)Google Scholar
  31. 31.
    Higman, G.: A finitely generated infinite simple group. J. Lond. Math. Soc. 26, 61–64 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Higman, G.: Finitely Presented Infinite Simple Groups. Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, 1974. Notes on Pure Mathematics, No. 8 (1974)Google Scholar
  33. 33.
    Kropholler, P.H., Mislin, G.: Groups acting on finite-dimensional spaces with finite stabilizers. Comment. Math. Helv. 73(1), 122–136 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kochloukova, D.H.: The \(\text{ FP }_m\)-conjecture for a class of metabelian groups. J. Algebra 184(3), 1175–1204 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Kropholler, P.H.: On groups of type \((\text{ FP })_\infty \). J. Pure Appl. Algebra 90(1), 55–67 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Kochloukova, D.H., Sidki, S.N.: Self-Similar Groups of Type \(FP_n\). arXiv:1710.04745
  37. 37.
    Le Boudec, A.: Compact presentability of tree almost automorphism groups. Ann. Inst. Fourier 67(1), 329–365 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Leary, I.J.: Subgroups of Almost Finitely Presented Groups. arXiv:1610.05813
  39. 39.
    Leary, I.J.: Uncountably Many Groups of Type \(FP\). arXiv:1512.06609
  40. 40.
    Meier, J., Meinert, H., VanWyk, L.: Higher generation subgroup sets and the \(\Sigma \)-invariants of graph groups. Comment. Math. Helv. 73(1), 22–44 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Martínez-Pérez, C., Matucci, F., Nucinkis, B.E.A.: Cohomological finiteness conditions and centralisers in generalisations of Thompson’s group \(V\). Forum Math. 28(5), 909–921 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Nekrashevych, V.V.: Cuntz–Pimsner algebras of group actions. J. Oper. Theory 52(2), 223–249 (2004)MathSciNetGoogle Scholar
  43. 43.
    Nekrashevych, V.: Self-Similar Groups, Mathematical Surveys and Monographs, vol. 117. American Mathematical Society, Providence (2005)CrossRefzbMATHGoogle Scholar
  44. 44.
    Nekrashevych, V.: Finitely presented groups associated with expanding maps. In: Geometric and Cohomological Group Theory London Mathematical Society Lecture Notes Series. Cambridge University Press, Cambridge (2017). arXiv:1312.5654
  45. 45.
    Niederreiter, H., Xing, C.: Algebraic Geometry in Coding Theory and Cryptography. Princeton University Press, Princeton (2009)zbMATHGoogle Scholar
  46. 46.
    Röver, C.E.: Constructing finitely presented simple groups that contain Grigorchuk groups. J. Algebra 220(1), 284–313 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Stallings, J.: A finitely presented group whose 3-dimensional integral homology is not finitely generated. Am. J. Math. 85, 541–543 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Stein, M.: Groups of piecewise linear homeomorphisms. Trans. Am. Math. Soc. 332(2), 477–514 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Skipper, R., Zaremsky, M.C.B.: Almost-automorphisms of Trees, Cloning Systems and Finiteness Properties. (submitted). arXiv:1709.06524
  50. 50.
    Thumann, W.: Operad groups and their finiteness properties. Adv. Math. 307, 417–487 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Weil, A.: Basic number theory. Die Grundlehren der mathematischen Wissenschaften, Band 144. Springer, New York (1967)Google Scholar
  52. 52.
    Witzel, S.: Classifying Spaces from Ore Categories with Garside Families. arXiv:1710.02992
  53. 53.
    Witzel, S.: Finiteness Properties of Arithmetic Groups Acting on Twin Buildings. Lecture Notes in Mathematics, vol. 2109. Springer, Cham (2014)Google Scholar
  54. 54.
    Witzel, S., Zaremsky, M.C.B.: The Basilica Thompson group is not finitely presented. Groups Geom. Dyn. (to appear). arXiv:1603.01150
  55. 55.
    Witzel, S., Zaremsky, M.C.B.: Thompson groups for systems of groups, and their finiteness properties. Groups Geom. Dyn. 12(1), 289–358 (2018)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Rachel Skipper
    • 1
    Email author
  • Stefan Witzel
    • 2
  • Matthew C. B. Zaremsky
    • 3
  1. 1.Mathematics InstituteUniversity of GöttingenGöttingenGermany
  2. 2.Department of MathematicsBielefeld UniversityBielefeldGermany
  3. 3.Department of Mathematics and StatisticsUniversity at Albany (SUNY)AlbanyUSA

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