Subdivision rules and a space at infinity for the n-dimensional torus

Details Subdivision rules and a space at infinity for the n-dimensional torus Subdivision rules and a space at infinity for the n-dimensional torus

2011-10-14

arxiv.org/abs/1110.3310v1

Mathematics

Geometric Topology

13 pages, 12 figures

Scientific paper

Cannon and Swenson have shown that all hyperbolic groups with a 2-sphere at infinity have an associated subdivision rule, and that this subdivision rule captures the action of hyperbolic 3-manifolds on hyperbolic 3-space. We extend this idea by finding subdivision rules on the (n-1)-sphere for the n-dimensional torus, showing that such structures exist in other dimensions and in other geometries. We define a topological space at infinity that generalizes the sphere at infinity of hyperbolic manifolds. The n-dimensional tori are particularly interesting, as their spaces at infinity have subsets that are Hausdorff only in certain directions.

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