Martin Lustig

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

Most automorphisms of a hyperbolic group have very simple dynamics

Abstract Let G be a non-elementary hyperbolic group (e.g. a non-abelian free group of finite rank). We show that, for “most” automorphisms α of G (in a precise sense), there exist distinct elements X+,X− in the Gromov boundary ∂G of G such that lim n→+∞ α ±n (g)=X ± for every g∈G which is not periodic under α . This follows from the fact that the homeomorphism ∂α induced on ∂G has North–South (loxodromic) dynamics.

IRREDUCIBLE AUTOMORPHISMS OF

We show that if an automorphism of a non-abelian free group

F_{n}

F_n

HAVE NORTH–SOUTH DYNAMICS ON COMPACTIFIED OUTER SPACE

is irreducible with irreducible powers, it acts on the boundary of Culler–Vogtmann’s outer space with north–south dynamics: there are two fixed points, one attracting and one repelling, and orbits accumulate only on these points. The main new tool we use is the equivariant assignment of a point

Closed incompressible surfaces in complements of wide knots and links

Q(X)

Automorphisms of free groups have asymptotically periodic dynamics

to any end

On the dynamics and the fixed subgroup of a free group automorphism

Xinpartial F_n

Manifolds with irreducible Heegaard splittings of high genus

, given an action of

Horizontal Dehn surgery and genericity in the curve complex

F_n