Involutions of negatively curved groups with wild boundary behavior

We are interested in examples of compact, complete, locally CAT(-1) spaces X, and closed totally geodesic codimension two subspaces Y , with the property that ∂∞X̃ = Sn+2 ∞ , and ∂ ∞Ỹ = Sn ∞. We show that if the inclusion S n ∞ ↪→ Sn+2 ∞ induced by the inclusion Y ⊂ X is knotted, then it is a totally wild knot (i.e. nowhere tame). We give examples where the inclusion Sn ∞ ↪→ Sn+2 ∞ is indeed knotted. Furthermore, the examples Y ⊂ X we construct can be realized as fixed point sets of involutive isometries of X, so that the corresponding totally wildly knotted Sn ∞ ↪→ Sn+2 ∞ are fixed point sets of geometric involutions on Sn+2 ∞ = ∂ ∞X̃. In the appendix, we also include a complete criterion for knottedness of tame codimension two spheres in high dimensional (≥ 6) spheres.