Marshall M. Cohen

Very small group actions on R-trees and dehn twist automorphisms

LENGTH functions on a group G which come from actions on R-trees [20,29,10,3,37] and spaces of such length functions have been of central importance in combinatorial group theory in recent years. We will be concerned with subspaces of the projective space SLF(G) of translation length functions of small actions of G on R-trees (see [ 121 or Section 1 below). An element of this space will often be referred to simply as “an action” (see 1.7). Thurston’s classification of diffeomorphisms of surfaces [36] and the work of Morgan and Shalen [21-231 has led to a well-known program for studying the structure of an individual automorphism of G or of the outer automorphism group Out(G). An appropriate subspace X of SLF(G) is taken as an analogue of Teichmuller space or its boundary, and Out(G) is viewed as an analogue of the mapping class group. One hopes to learn about Out(G) through its induced action on the space of actions X [13, 9, 19, 18, 241 and to analyze an individual automorphism by finding fixed points and studying the dynamics of its induced action on X [S, 191. For the free group of rank it, an important underpinning, the contractibility of the spaces of actions in question, has been proven [34,35]. See [l, 12,30, 311 for more complete references and history. We contribute to this program in several ways. We start by introducing (Section 2) the space VSL(G)-the projective space of translation length functions of very small actions of G on R-trees. A very small action of the group G on an R-tree Y is a small action such that for each non-trivial g E G the fixed subtree Fix(g) (a) is equal to Fix(g”) whenever g” # 1 and (b) does not contain a triod (cone on three points). Notice that if Free(G) denotes the space of free actions of G on R-trees and SLF(G) denotes the space of small actions [12] then

The surjectivity problem for one-generator, one-relator extensions of torsion-free groups

We prove that the natural map G! b G ,w here Gis a torsion-free group and b G is obtained by adding a new generator t and a new relator w, is surjective only if w is conjugate to gt or gt 1 where g2 G. This solves a special case of

A proof of Newman's theorem

If Σ n is a piecewise linear n-sphere and Q n is a piecewise linear n-ball which is a subpolyhedron of Σ n then is a piecewise linear n-ball .

A Geometric Approach to Homotopy Theory

From here on all CW complexes mentioned will be assumed finite unless they occur as the covering spaces of given finite complexes.

Whitehead Torsion in the CW Category

The geometry in Chapter II and the algebraic analysis of Chapter III are synthesized in the definition: If (K, L) is a pair of finite, connected CW complexes such that K ↯ L then the torsion of (K, L)—written τ(K, L)—is defined by

Simplicial Structures and Transverse Cellularity

\tau (K,L) = \tau (C(\tilde K,\tilde L)) \in Wh({\pi _1}L)

PATHS OF GEODESICS AND GEOMETRIC INTERSECTION NUMBERS: II

where (K, L) is the universal covering of (K,L).