LLP for surface groups and hyperbolic 3-manifold groups

Establish whether the maximal group C*-algebra C*(Γ) of Γ equal to a surface group or to the fundamental group of a closed hyperbolic three-manifold satisfies Kirchberg’s local lifting property (LLP), so that unit ball–valued quasi-representations can be uniformly approximated by ucp quasi-representations and the ucp hypothesis can be removed from the conditional weak stability results proved in the paper.

Background

The local lifting property (LLP) is a central technical tool in the paper: it enables approximation of general quasi-representations by unital completely positive (ucp) quasi-representations, which is needed to apply the new stable uniqueness theorem. The authors show their main conditional weak stability results under a ucp assumption whenever LLP is unavailable.

LLP is known for amenable groups and for some constructions (e.g., certain free products and semidirect products with amenable groups), but its status for many geometrically significant groups is unclear. If surface groups or hyperbolic 3–manifold groups had LLP, then the ucp restriction could be removed from several main theorems. The authors explicitly conjecture LLP holds in these cases.