On aspherical presentations of groups

Abstract

The Whitehead asphericity conjecture claims that if ⟨ A ‖ R ⟩ \langle \, \mathcal {A} \, \| \, \mathcal {R} \, \rangle is an aspherical group presentation, then for every S ⊂ R \mathcal {S} \subset \mathcal {R} the subpresentation ⟨ A ‖ S ⟩ \langle \, \mathcal {A} \, \| \, \mathcal {S} \, \rangle is also aspherical. This conjecture is generalized for presentations of groups with periodic elements by introduction of almost aspherical presentations. It is proven that the generalized Whitehead asphericity conjecture (which claims that every subpresentation of an almost aspherical presentation is also almost aspherical) is equivalent to the original Whitehead conjecture and holds for standard presentations of free Burnside groups of large odd exponent, Tarski monsters and some others. Next, it is proven that if the Whitehead conjecture is false, then there is an aspherical presentation E = ⟨ A ‖ R ∪ z ⟩ E = \langle \, \mathcal {A} \, \| \, \mathcal {R} \cup z \, \rangle of the trivial group E E , where the alphabet A \mathcal {A} is finite or countably infinite and z ∈ A z \in \mathcal {A} , such that its subpresentation ⟨ A ‖ R ⟩ \langle \, \mathcal {A} \, \| \, \mathcal {R} \, \rangle is not aspherical. It is also proven that if the Whitehead conjecture fails for finite presentations (i.e., with finite A \mathcal {A} and R \mathcal {R} ), then there is a finite aspherical presentation ⟨ A ‖ R ⟩ \langle \, \mathcal {A} \, \| \, \mathcal {R} \, \rangle , R = { R 1 , R 2 , … , R n } \mathcal {R} = \{ R_{1}, R_{2}, \dots , R_{n} \} , such that for every S ⊆ R \mathcal {S} \subseteq \mathcal {R} the subpresentation ⟨ A ‖ S ⟩ \langle \, \mathcal {A} \, \| \, \mathcal {S} \, \rangle is aspherical and the subpresentation ⟨ A ‖ R 1 R 2 , R 3 , … , R n ⟩ \langle \, \mathcal {A} \, \| \, R_{1}R_{2}, R_{3}, \dots , R_{n}\, \rangle of aspherical ⟨ A ‖ R 1 R 2 , R 2 , R 3 , … , R n ⟩ \langle \, \mathcal {A} \, \| \, R_{1}R_{2}, R_{2}, R_{3}, \dots , R_{n}\, \rangle is not aspherical. Now suppose a group presentation H = ⟨ A ‖ R ⟩ H = \langle \, \mathcal {A} \, \| \, \mathcal {R} \, \rangle is aspherical, x ∉ A x \not \in \mathcal {A} , W ( A ∪ x ) W(\mathcal {A} \cup x) is a word in the alphabet ( A ∪ x ) ± 1 (\mathcal {A} \cup x)^{\pm 1} with nonzero sum of exponents on x x , and the group H H naturally embeds in G = ⟨ A ∪ x ‖ R ∪ W ( A ∪ x ) ⟩ G = \langle \, \mathcal {A} \cup x \, \| \, \mathcal {R} \cup W(\mathcal {A} \cup x) \, \rangle . It is conjectured that the presentation G = ⟨ A ∪ x ‖ R ∪ W ( A ∪ x ) ⟩ G = \langle \, \mathcal {A} \cup x \, \| \, \mathcal {R} \cup W(\mathcal {A} \cup x) \, \rangle is aspherical if and only if G G is torsion free. It is proven that if this conjecture is false and G = ⟨ A ∪ x ‖ R ∪ W ( A ∪ x ) ⟩ G = \langle \, \mathcal {A} \cup x \, \| \, \mathcal {R} \cup W(\mathcal {A} \cup x) \, \rangle is a counterexample, then the integral group ring Z ( G ) \mathbb {Z}(G) of the torsion free group G G will contain zero divisors. Some special cases where this conjecture holds are also indicated.