Complex hyperbolic 2-orbifolds with isolated singularities

<jats:p> For each prime <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0017089525100773_inline1.png"/> <jats:tex-math>$p$</jats:tex-math> </jats:alternatives> </jats:inline-formula> , this paper constructs compact complex hyperbolic <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0017089525100773_inline2.png"/> <jats:tex-math>$2$</jats:tex-math> </jats:alternatives> </jats:inline-formula> -manifolds with an isometric action of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0017089525100773_inline3.png"/> <jats:tex-math>$\mathbb{Z} / p \mathbb{Z}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> that is not free and has only isolated fixed points. The case <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0017089525100773_inline4.png"/> <jats:tex-math>$p = 2$</jats:tex-math> </jats:alternatives> </jats:inline-formula> is special, and finding general examples for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0017089525100773_inline5.png"/> <jats:tex-math>$p=2$</jats:tex-math> </jats:alternatives> </jats:inline-formula> is related to whether or not complex hyperbolic lattices are conjugacy separable on torsion. </jats:p>