On aspherical presentations of groups

Ivanov, Sergei¹

¹ Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801

Electronic research announcements of the American Mathematical Society, Tome 04 (1998), pp. 109-114.

Résumé

The Whitehead asphericity conjecture claims that if $\langle \mathcal {A} \| \mathcal {R} \rangle$ is an aspherical group presentation, then for every $\mathcal {S} \subset \mathcal {R}$ the subpresentation $\langle \mathcal {A} \| \mathcal {S} \rangle$ 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 $E = \langle \mathcal {A} \| \mathcal {R} \cup z \rangle$ of the trivial group $E$, where the alphabet $\mathcal {A}$ is finite or countably infinite and $z \in \mathcal {A}$, such that its subpresentation $\langle \mathcal {A} \| \mathcal {R} \rangle$ is not aspherical. It is also proven that if the Whitehead conjecture fails for finite presentations (i.e., with finite $\mathcal {A}$ and $\mathcal {R}$), then there is a finite aspherical presentation $\langle \mathcal {A} \| \mathcal {R} \rangle$, $\mathcal {R} = \{ R_{1}, R_{2}, \dots , R_{n} \}$, such that for every $\mathcal {S} \subseteq \mathcal {R}$ the subpresentation $\langle \mathcal {A} \| \mathcal {S} \rangle$ is aspherical and the subpresentation $\langle \mathcal {A} \| R_{1}R_{2}, R_{3}, \dots , R_{n} \rangle$ of aspherical $\langle \mathcal {A} \| R_{1}R_{2}, R_{2}, R_{3}, \dots , R_{n} \rangle$ is not aspherical. Now suppose a group presentation $H = \langle \mathcal {A} \| \mathcal {R} \rangle$ is aspherical, $x \not \in \mathcal {A}$, $W(\mathcal {A} \cup x)$ is a word in the alphabet $(\mathcal {A} \cup x)^{\pm 1}$ with nonzero sum of exponents on $x$, and the group $H$ naturally embeds in $G = \langle \mathcal {A} \cup x \| \mathcal {R} \cup W(\mathcal {A} \cup x) \rangle$. It is conjectured that the presentation $G = \langle \mathcal {A} \cup x \| \mathcal {R} \cup W(\mathcal {A} \cup x) \rangle$ is aspherical if and only if $G$ is torsion free. It is proven that if this conjecture is false and $G = \langle \mathcal {A} \cup x \| \mathcal {R} \cup W(\mathcal {A} \cup x) \rangle$ is a counterexample, then the integral group ring $\mathbb {Z}(G)$ of the torsion free group $G$ will contain zero divisors. Some special cases where this conjecture holds are also indicated.

Détail

DOI : 10.1090/S1079-6762-98-00052-3

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