Groups of small homological dimension and the Atiyah Conjecture
نویسندگان
چکیده
A group has homological dimension ≤ 1 if it is locally free. We prove the converse provided that G satisfies the Atiyah Conjecture about L -Betti numbers. We also show that a finitely generated elementary amenable group G of cohomological dimension ≤ 2 possesses a finite 2dimensional model for BG and in particular that G is finitely presented and the trivial ZG-module Z has a 2-dimensional resolution by finitely generated free ZG-modules.
منابع مشابه
A counterexample to a question of Atiyah
We prove that there are examples of finitely generated groups Γ together with group ring elements Q ∈ QΓ for which the von Neumann dimension dimLΓ kerQ is irrational, so (in conjunction with other known results) disproving a conjecture of Atiyah.
متن کاملAn Etale Approach to the Novikov Conjecture
We show that the rational Novikov conjecture for a group Γ of finite homological type follows from the mod 2 acyclicity of the Higson compactifcation of an EΓ. We then show that for groups of finite asymptotic dimension the Higson compactification is mod p acyclic for all p, and deduce the integral Novikov conjecture for these groups.
متن کاملGraphs of groups and the Atiyah conjecture for one-relator groups
For a finitely-presented, torsion-free, discrete group G, the Atiyah conjecture asserts that the L-Betti numbers of any finite CW-complex with fundamental group G are integers; this conjecture has a natural extension to all groups. We prove that the class of groups for which the (extended) Atiyah conjecture holds and the finite subgroups have only finitely many different orders, is closed under...
متن کاملCopresented Dimension of Modules
In this paper, a new homological dimension of modules, copresented dimension, is defined. We study some basic properties of this homological dimension. Some ring extensions are considered, too. For instance, we prove that if $Sgeq R$ is a finite normalizing extension and $S_R$ is a projective module, then for each right $S$-module $M_S$, the copresented dimension of $M_S$ does not exceed the c...
متن کاملGorenstein homological dimensions with respect to a semi-dualizing module over group rings
Let R be a commutative noetherian ring and Γ a finite group. In this paper,we study Gorenstein homological dimensions of modules with respect to a semi-dualizing module over the group ring . It is shown that Gorenstein homological dimensions of an -RΓ module M with respect to a semi-dualizing module, are equal over R and RΓ .
متن کامل