Algebraic & Geometric Topology

$L^2$–invisibility of symmetric operad groups

Werner Thumann

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


We show a homological result for the class of planar or symmetric operad groups: under certain conditions, group (co)homology of such groups with certain coefficients vanishes in all dimensions, provided it vanishes in dimension 0. This can be applied, for example, to l2–homology or cohomology with coefficients in the group ring. As a corollary, we obtain explicit vanishing results for Thompson-like groups such as the Brin–Thompson groups nV .

Article information

Algebr. Geom. Topol., Volume 16, Number 4 (2016), 2229-2255.

Received: 21 July 2015
Accepted: 30 December 2015
First available in Project Euclid: 28 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20J05: Homological methods in group theory
Secondary: 22D10: Unitary representations of locally compact groups 18D50: Operads [See also 55P48]

operad groups Thompson groups group homology L2-homology


Thumann, Werner. $L^2$–invisibility of symmetric operad groups. Algebr. Geom. Topol. 16 (2016), no. 4, 2229--2255. doi:10.2140/agt.2016.16.2229.

Export citation


  • M,G Brin, Higher dimensional Thompson groups, Geom. Dedicata 108 (2004) 163–192
  • K,S Brown, Finiteness properties of groups, from: “Proceedings of the Northwestern conference on cohomology of groups”, J. Pure Appl. Algebra 44 (1987) 45–75
  • K,S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer, New York (1994)
  • K,S Brown, R Geoghegan, An infinite-dimensional torsion-free ${\rm FP}\sb{\infty }$ group, Invent. Math. 77 (1984) 367–381
  • M,W Davis, Poincaré duality groups, from: “Surveys on surgery theory, Vol 1”, (S Cappell, A Ranicki, J Rosenberg, editors), Ann. of Math. Stud. 145, Princeton Univ. Press (2000) 167–193
  • D,S Farley, B Hughes, Finiteness properties of some groups of local similarities, Proc. Edinb. Math. Soc. 58 (2015) 379–402
  • M Gromov, Large Riemannian manifolds, from: “Curvature and topology of Riemannian manifolds”, (K Shiohama, T Sakai, T Sunada, editors), Lecture Notes in Math. 1201, Springer, Berlin (1986) 108–121
  • W Lück, $L^2$–invariants: theory and applications to geometry and $K$–theory, Ergeb. Math. Grenzgeb. 44, Springer, Berlin (2002)
  • R Sauer, W Thumann, $l\sp 2$–invisibility and a class of local similarity groups, Compos. Math. 150 (2014) 1742–1754
  • W Thumann, Operad groups and their finiteness properties, preprint (2014)