We generalize a result of Moon on the fibering of certain 3-manifolds over
the circle. Our main theorem is the following: Let $M$ be a closed 3-manifold.
Suppose that $G=\pi_1(M)$ contains a finitely generated group $U$ of infinite
index in $G$ which contains a non-trivial subnormal subgroup $N\neq \mathbb{Z}$
of $G$, and suppose that $N$ has a composition series of length $n$ in which at
least $n-1$ terms are finitely generated. Suppose that $N$ intersects
nontrivially the fundamental groups… Expand

Schreier proved in [8] tha t a finitely generated normal subgroup U # { 1 } of a free group F is of finite index. This result was extended by Karrass and Solitar in [3], to the case when U is not… Expand

This book describes the global properties of simply-connected spaces that are non-positively curved in the sense of A. D. Alexandrov, and the structure of groups which act on such spaces by… Expand

The theory of 3-manifolds has been revolutionised in the last few years by work of Thurston [66-70]. He has shown that geometry has an important role to play in the theory in addition to the use of… Expand

Fibre bundles, an integral part of differential geometry, are also important to physics. This text, a succint introduction to fibre bundles, includes such topics as differentiable manifolds and… Expand

We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric… Expand