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The incipient infinite cluster in high-dimensional percolation
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On aspherical presentations of groups
Sergei V. Ivanov
Abstract.
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.
Copyright 1998 American Mathematical Society
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Article Info
- ERA Amer. Math. Soc. 04 (1998), pp. 109-114
- Publisher Identifier: S 1079-6762(98)00052-3
- 1991 Mathematics Subject Classification. Primary 20F05, 20F06, 20F32
- Key words and phrases
- Received by the editors April 13, 1998
- Posted on December 15, 1998
- Communicated by Efim Zelmanov
- Comments (When Available)
Sergei V. Ivanov
Department of Mathematics,
University of Illinois at Urbana-Champaign,
1409 West Green Street, Urbana,
IL 61801
E-mail address: ivanov@math.uiuc.edu
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