On aspherical presentations of groups
Authors: Ivanov Sergei 
Published: July 2002

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.