Existence of singular discrete isometry groups for CAT(0) spaces with Lie isometry group

Determine whether there exist proper, geodesically complete CAT(0) spaces X for which the full isometry group Isom(X) is a Lie group and that admit a singular discrete subgroup Γ < Isom(X), i.e., a discrete group of isometries whose action on X has no point with trivial stabilizer.

Background

The paper establishes strong splitting results for uniform lattices acting on proper, geodesically complete CAT(0) spaces whose full isometry group is Lie, including virtual product decompositions and lower bounds on diastole. In contrast to general CAT(0) settings, these Lie-group contexts enable precise control over the limit behavior under collapsing and convergence.

Despite these structural results, the authors note a gap in examples: while many properties are known for spaces with Lie isometry groups, it remains unknown whether such spaces can admit singular discrete isometry groups (actions with no point having trivial stabilizer). This existence question touches the boundaries between Lie-group symmetry and the nature of discrete group actions with torsion or fixed-point phenomena in CAT(0) geometry.