Introduction to Group Actions on Trees

  • Michael Kapovich
Part of the Modern Birkhäuser Classics book series (MBC)


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.


Fundamental Group Geodesic Segment Isometric Action Geodesic Triangle Axial Element 
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Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, DavisDavisU.S.A.

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