Hyperbolic Manifolds and Discrete Groups pp 227-241 | Cite as

# Introduction to Group Actions on Trees

## Abstract

Recall that a metric tree *T* is a metrically complete nonempty geodesic metric space where each geodesic triangle is a tripod. In particular, every two points in *T* are connected by unique geodesic segment. *Vertices* (or *branch points*) of *T* are the points *x* ϵ *T* such that there are at least three geodesic segments in *T* emanating from *x* whose interiors are disjoint. A *nondegenerate* tree is a tree that contains more than one point. We shall consider isometric actions of groups *G* on trees *T*. The pair of a tree *T* and an action of *G* on *T* is called a *G*-tree. A *G*-tree is called *minimal* if there is no *-G*-invariant subtree in *T* differenct from *T*. A *G*-tree is called *trivial* in this case.) The action is called *small* if the stabilizer of any nondegenerate arc is a virtually nilpotent subgroup of *G*. In this case, we will say that the *G*-tree *T* is *small*. Two *G*-trees are said to be *isomorphic* if there is a *G*-equivariant isometry between therse trees.

## Keywords

Fundamental Group Geodesic Segment Isometric Action Geodesic Triangle Axial Element## Preview

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