Algebraic & Geometric Topology

Euler characteristics and actions of automorphism groups of free groups

Shengkui Ye

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let M r be a connected orientable manifold with the Euler characteristic χ ( M ) 0 mod 6 . Denote by SAut ( F n ) the unique subgroup of index two in the automorphism group of a free group. Then any group action of SAut ( F n ) (and thus the special linear group SL n ( ) ) with n r + 2 on M r by homeomorphisms is trivial. This confirms a conjecture related to Zimmer’s program for these manifolds.

Article information

Algebr. Geom. Topol., Volume 18, Number 2 (2018), 1195-1204.

Received: 15 August 2017
Revised: 25 September 2017
Accepted: 5 October 2017
First available in Project Euclid: 22 March 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57S20: Noncompact Lie groups of transformations
Secondary: 57S17: Finite transformation groups

Zimmer's program Euler characteristics matrix group actions


Ye, Shengkui. Euler characteristics and actions of automorphism groups of free groups. Algebr. Geom. Topol. 18 (2018), no. 2, 1195--1204. doi:10.2140/agt.2018.18.1195.

Export citation


  • M Belolipetsky, A Lubotzky, Finite groups and hyperbolic manifolds, Invent. Math. 162 (2005) 459–472
  • A Borel, Seminar on transformation groups, Annals of Mathematics Studies 46, Princeton Univ. Press (1960)
  • G E Bredon, Orientation in generalized manifolds and applications to the theory of transformation groups, Michigan Math. J. 7 (1960) 35–64
  • G E Bredon, Sheaf theory, 2nd edition, Graduate Texts in Mathematics 170, Springer (1997)
  • M R Bridson, K Vogtmann, Actions of automorphism groups of free groups on homology spheres and acyclic manifolds, Comment. Math. Helv. 86 (2011) 73–90
  • A Brown, D Fisher, S Hurtado, Zimmer's conjecture: subexponential growth, measure rigidity, and strong property $(T)$, preprint (2016)
  • A Brown, F R Hertz, Z Wang, Invariant measures and measurable projective factors for actions of higher-rank lattices on manifolds, preprint (2016)
  • B Farb, P Shalen, Real-analytic actions of lattices, Invent. Math. 135 (1999) 273–296
  • D Fisher, Groups acting on manifolds: around the Zimmer program, from “Geometry, rigidity, and group actions” (B Farb, D Fisher, editors), Univ. Chicago Press (2011) 72–157
  • S M Gersten, A presentation for the special automorphism group of a free group, J. Pure Appl. Algebra 33 (1984) 269–279
  • F Grunewald, A Lubotzky, Linear representations of the automorphism group of a free group, Geom. Funct. Anal. 18 (2009) 1564–1608
  • R S Kulkarni, Symmetries of surfaces, Topology 26 (1987) 195–203
  • L N Mann, J C Su, Actions of elementary $p$–groups on manifolds, Trans. Amer. Math. Soc. 106 (1963) 115–126
  • S Weinberger, ${\rm SL}(n,\mathbf Z)$ cannot act on small tori, from “Geometric topology” (W H Kazez, editor), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc., Providence, RI (1997) 406–408
  • D Witte, Arithmetic groups of higher ${\bf Q}$–rank cannot act on $1$–manifolds, Proc. Amer. Math. Soc. 122 (1994) 333–340
  • S Ye, Symmetries of flat manifolds, Jordan property and the general Zimmer program, preprint (2017)
  • R J Zimmer, D W Morris, Ergodic theory, groups, and geometry, CBMS Regional Conference Series in Mathematics 109, Amer. Math. Soc., Providence, RI (2008)