Groups of Cohomological Dimension One
- PDF / 4,119,562 Bytes
- 105 Pages / 504 x 720 pts Page_size
- 28 Downloads / 235 Views
245 Daniel E. Cohen Queen Mary College, London/G. B.
Groups of Cohomological Dimension One
Springer-Verlag Berlin' Heidelberg· New York 1972
AMS Subject Classifications (1970): 16 A 26, 20J 05
ISBN 3-540-05759-5 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-05759-5 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations. broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin' Heidelberg 1972. Library of Congress Catalog Card Number 71-189311. Printed in Germany. Offsetdruck: Julius Beltz, Hernsbach/Bergstr,
INTRODUCTION
Free groups have cohomological dimension one, so it is natural to ask whether the converse holds.
This question became of extra interest after it was shown that
the similar result holds for pro-p groups.
In 1968 Stallings [ 17 J showed that finitely generated groups of cohomological dimension one are free, and in 1969 Swan [19], using Stallings' work, solved the general problem. THEOREM A
A group of cohomological dimension one (over some ring with unit)
is free (provided it is torsi on-free ). Stall ings and Swan also proved another theorem with an analogue for pro-p groups. THEOREM B
A torsion-free group containing a free subgroup of finite index is free. This follows immediately from Theorem A and the following result of Serre.
THEOREM C
Let
of finite index. If G
R be a commutative ring with unity, G a group with a subgroup H
has no R-torsion (e.9.
if
have the same cohomological dimension over
R.
G
is torsion-free) then
G
and
H
These notes, based on lectures given at King's College, London, give a completely self-contained account of these theorems.
An elementary knowledge of
combinatorial group theory and homological algebra is needed, but the theorems of Kuros' and GruSko on free products are proved. The notes differ from the papers of Stallings and Swan in several significant details, among them the following: i)
the theory of ends is given in the algebraic form due to the author [ 2] i
ii)
a key lemma for Stallings' structure theorem for groups with infinitely many
ends is proved by Dunwoody's method [ 3 Ji iii)
this structure theorem is given the proof recently obtained by a research student
IV at Queen Mary College; iv)
some of Swan's homological arguments are replaced by more explicit discussion
of the augmentation ideal Theorem A
v)
THEOREM D factor of
Let
G iff
IG of a group
G;
is relativised to give a result implying the following theorem; H
be a subgroup of a free group
is a summand of
I
G
G.
Then
H
is a free
.
My thanks are due to C. R. Leedham-Green for his careful reading of these note
Data Loading...
