Hyperbolic Manifolds and Discrete Groups pp 31-56 | Cite as

# Geometry of Hyperbolic Space

## Abstract

If *G* is a group acting on a set *X*, then for each *g ϵ G* we let Fix *X*(*g*) denote the fixed-point set for the action of *g* on *X* and Fix*X*(*G*) the subset of *X* fixed by the whole group *G* (pointwise). Sometimes we shall omit the subscript *X* in this notation when it is clear what the space *X* is. If *Y ⊂ X* is a subset, then we will use the notation Fix*G*(*Y*) to denote the collection of elements *g ϵ G* that fix *Y* pointwise. A sequence *x* _{ n } in a topological space *X* (i.e., each subsequence in {*x* _{ n }} contains a convergent subsequence). We will say that *x* _{ n } *subconverges* to *x ϵ X* if *x* is the limit of a subsequence in {*x* _{ n }}.

## Keywords

Hyperbolic Space Cayley Graph Hausdorff Distance Geodesic Segment Ideal Boundary## Preview

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