1,721,018 research outputs found
The integral cohomology rings of some p-groups
No full textWe determine the integral cohomology rings of an infinite family of p-groups, for odd primes p, with cyclic derived subgroups. Our method involves embedding the groups in a compact Lie group of dimension one, and was suggested independently by P. H. Kropholler and J. Huebschmann. This construction has also been used by the author to calculate the mod-p cohomology of the same groups and by B. Moselle to obtain partial results concerning the mod-p cohomology of the extra special p-group
A differential in the Lyndon—Hochschild—Serre spectral sequence
No full textWe consider the Lyndon-Hochschild-Serre spectral sequence with coefficients in the field of p elements for central extensions in which the kernel is cyclic of order a power of p. For these spectral sequences the second and third differentials are known; we give a description of the fourth differential. The differential from odd rows to even rows involves a Massey triple product, and we calculate these products in the cohomology of any finite abelian group. <br/
p-Groups are not determined by their integral cohomology groups
No full textFor each prime p we exhibit non-isomorphic p-groups that have isomorphic integral cohomology groups. We give an indirect proof rather than a computation. The examples for p=2 in the original paper were flawed; an erratum repaired this
3-groups are not determined by their integral cohomology rings
No full textThere is one compact 1-dimensional Lie group with 27 components and nilpotence class three. We give a presentation for the integral cohomology ring of this group. We show that finite groups of order 81 can be distinguished by their first few integral cohomology groups, and we exhibit a pair of groups of order 243 that have isomorphic integral cohomology rings. <br/
A note on torsion length and torsion subgroups
Get PDFAnswering Questions 19.23 and 19.24 from the Kourovka notebook we construct
polycyclic groups with arbitrary torsion lengths and give examples of finitely
presented groups whose quotients by their torsion subgroups are not finitely
presented.Comment: 4 pages Second version: 6 pages, more theory include
Equivariant vector bundles over classifying spaces for proper actions
No full textWe give the first examples of discrete groups G for which there exist compatible collections of representations over the finite subgroups of G that do not come from any G-equivariant (virtual) vector bundle on the classifying space for proper actions of G. This implies that the Atiyah-Hirzebruch spectral sequence for computing the equivariant topological K-theory of the classifying space for proper actions of G has non-zero differentials. We show that this cannot happen for right-angled Coxeter groups, and we compute K-theory for these groups
A metric Kan-Thurston theorem
No full textFor every simplicial complex X we construct a locally CAT(0) cubical complex T(X), a cellular isometric involution tau on T(X) and a map t from T(X) to X with the following properties: t is equivariant for tau; t is a homology isomorphism; the induced map from the quotient space T(X)/tau to X is a homotopy equivalence; the induced map from the tau-fixed point set in T(X) to X is a homology isomorphism. The construction is functorial in X. One corollary is an equivariant Kan-Thurston theorem: every connected proper G-CW-complex has the same equivariant homology as the classifying space for proper actions of some other group. From this we obtain extensions of a theorem of Quillen on the spectrum of a (Borel) equivariant cohomology ring and of a result of Block concerning assembly conjectures. Another corollary of our main result is that there can be no algorithm to decide whether a CAT(0) cubical group is generated by torsion. In appendices we prove some foundational results concerning cubical complexes, notably in the infinite-dimensional case. We characterize the cubical complexes for which the natural metric is complete; we establish Gromov's criterion for infinite-dimensional cubical complexes; we show that every CAT(0) cube complex is cubical; we deduce that the second cubical subdivision of any locally CAT(0) cube complex is cubical. [A version of this paper was submitted in September 2010. This is a revised version I made in April 2011 (improvements to some material in the appendices).]<br/
Commensurating HNN-extensions: non-positive curvature and biautomaticity
No full textWe show that the commensurator of any quasiconvex abelian subgroup in a biautomatic group is small, in the sense that it has finite image in the abstract commensurator of the subgroup. Using this criterion we exhibit groups that are CAT(0) but not biautomatic. These groups also resolve a number of other questions concerning CAT(0) groups
Groups of type <i>FP</i> via graphical small cancellation
No full textWe construct an uncountable family of groups of type FP. In contrast to every previous construction of non-finitely presented groups of type FP we do not use Morse theory on cubical complexes; instead we use Gromov's graphical small cancellation. <br/
The Yagita invariant of symplectic groups of large rank
No full textFix a prime p and let R be a subring of the complex numbers that is either integrally closed or contains a primitive pth root of 1. For any such R and any n greater than or equal to p-1, we compute the Yagita invariant at the prime p for the symplectic group Sp(2n,R). <br/
- …
