Parabolic isometry of hyperbolic three-space

A parabolic isometry of hyperbolic three-space is a nonidentity orientation-preserving isometry that fixes exactly one point of the ideal boundary sphere and no point in the interior.

Normal form

Under the PSL₂(ℂ) action, it is conjugate to

z\longmapsto z+1, \qquad (z,r)\longmapsto(z+1,r).

The corresponding matrix is $\begin{pmatrix}1&1\\0&1\end{pmatrix}$ , considered modulo sign. It fixes $\infty$ and preserves every horizontal horosphere.

Matrix test

For a lift $A\in\operatorname{SL}_2(\mathbb C)$ , the test is $(\operatorname{tr} A)^2=4$ and $A\ne\pm I$ . The scalar exclusion prevents misclassifying the identity.

References

F. W. Gehring, C. Maclachlan, G. J. Martin, and A. W. Reid, Arithmeticity, discreteness and volume, Transactions of the AMS 349 (1997). Author-hosted paper, §2, classification of elements.