Conditional representation stability holds in a weak sense for fundamental groups of closed surfaces, certain Baumslag-Solitar groups, free-by-cyclic groups, and many fundamental groups of three-manifolds.
On the (Local) Lifting Property
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abstract
The (Local) Lifting Property ((L)LP) is introduced by Kirchberg and deals with lifting completely positive maps. We give a characterization of the (L)LP in terms of lifting $\ast$-homomorphisms. We use it to prove that if $A$ and $B$ have the LP and $F$ is their finite-dimensional C*-subalgebra, then $A\ast_F B$ has the LP. This answers a question of Ozawa. We prove that Exel's soft tori have the LP. As a consequence we obtain that $C^*(F_n\times F_n)$ is inductive limit of RFD C*-algebras with the LP. We prove that for a class of C*-algebras including $C^*(F_n\times F_n)$, all contractible C*-algebras and all suspensions, the LLP is equivalent to Ext being a group. As byproduct of methods developed in the paper we generalize Kirchberg's theorem about extensions with the WEP, give short proofs of several, old and new, facts about soft tori, new unified proofs of Li and Shen's characterization of RFD property of free products amalgamated over a finite-dimensional subalgebra and Blackadar's characterization of semiprojectivity of them.
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3-manifold groups, limit groups, and selected one-relator and right-angled Artin groups possess the local lifting property and property FD, implying flexible stability of their approximate representations.
citing papers explorer
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Conditional representation stability, classification of $*$-homomorphisms, and relative eta invariants
Conditional representation stability holds in a weak sense for fundamental groups of closed surfaces, certain Baumslag-Solitar groups, free-by-cyclic groups, and many fundamental groups of three-manifolds.
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The Local Lifting Property, Property FD, and stability of approximate representations
3-manifold groups, limit groups, and selected one-relator and right-angled Artin groups possess the local lifting property and property FD, implying flexible stability of their approximate representations.