Abstract
The main theorem here gives a geometric condition on the fixed point sets of two hyperbolic isometries of a CAT(0) group which guarantees that the subgroup generated by the two elements contains a free subgroup. A result in section 3 shows that an element extends to the identity on the boundary if and only if the element is virtually central in the CAT(0) group. Finally, the two dimensional case is considered in the last section of the paper and there it is shown that powers of two hyperbolic isometries commute if and only if a geometric condition on the fixed point sets is satisfied.
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Ruane, K.E. Dynamics of the Action of a CAT(0) Group on the Boundary. Geometriae Dedicata 84, 81–99 (2001). https://doi.org/10.1023/A:1010301824765
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DOI: https://doi.org/10.1023/A:1010301824765