4 D ec 1 99 7 CANONICAL DECOMPOSITIONS OF 3 - MANIFOLDS

Canonical Decompositions of 3–Manifolds

We describe a new approach to the canonical decompositions of 3–manifolds along tori and annuli due to Jaco–Shalen and Johannson (with ideas from Waldhausen) — the so-called JSJ–decomposition theorem. This approach gives an accessible proof of the decomposition theorem; in particular it does not use the annulus–torus theorems, and the theory of Seifert fibrations does not need to be developed i...

D ec 1 99 7 11 . 5 GENUS TWO HEEGAARD SPLITTINGS OF ORIENTABLE THREE - MANIFOLDS

Infinite-dimensional versions of the primary, cyclic and Jordan decompositions

The famous primary and cyclic decomposition theorems along with the tightly related rational and Jordan canonical forms are extended to linear spaces of infinite dimensions with counterexamples showing the scope of extensions.

D ec 1 99 5 On a universal quantum invariant of 3 - manifolds ∗

2 00 1 Regular Neighbourhoods and Canonical Decompositions for Groups ( Preliminary

We find canonical decompositions for finitely presented groups which specialize to the classical JSJ-decomposition when restricted to the fundamental groups of Haken manifolds. The decompositions that we obtain are invariant under automorphisms of the group. A crucial new ingredient is the concept of a regular neighbourhood for a finite family of almost invariant subsets of a group. An almost i...