Remarks on approximability and stability for groups

In this paper, we provide several instances in which interesting approximation and stability properties are inherited by quotients with respect to finitely generated normal subgroups or, more strongly, normal subgroups with Kazhdan’s property (T). Applications arise when these observations are combined with variations of the Rips construction due to Wise and Belegradek–Osin.

💡 Research Summary

The paper investigates how several important approximation and stability properties of countable groups behave under quotients by normal subgroups that are either finitely generated or have Kazhdan’s property (T). The authors focus on four main notions: soficity (the existence of asymptotic permutation models separating the kernel of a presentation), flexible P‑stability (the ability to correct asymptotic homomorphisms into genuine homomorphisms after adding a vanishing proportion of extra points), stability in finite actions (the requirement that any sofic approximation has a limit action weakly contained in the class of finite actions), and residual finite‑dimensionality (RFD) of the full group C*-algebra (the existence of enough finite‑dimensional *‑representations to separate points).

The central result, called Theorem A, states that for a short exact sequence
(1\to N\to G\stackrel{\pi}{\to} Q\to1)
the following hold:

1. If (N) is finitely generated and (G) is flexibly P‑stable, then the quotient (Q) is also flexibly P‑stable.
2. If (N) has property (T), (G) is residually finite, stable in finite actions, and (Q) is sofic, then (Q) is residually finite.
3. If (N) has property (T) and (G) is RFD, then (Q) is RFD.
4. If (N) has property (T) and the canonical p.m.p. action (G\curvearrowright{0,1}^Q) is sofic, then (Q) is sofic.

The proofs are spread over Sections 2–6. In Section 2 the authors recall the Hamming metric on symmetric groups and prove a key combinatorial lemma: if a finite set of permutations each moves at most an (\varepsilon)‑fraction of points, then any word in those permutations moves at most (m\varepsilon) points, where (m) is the number of generators. Using this, they show that when (N) is finitely generated, the almost‑trivial action of the generators on a large subset can be turned into an actual trivial action on a slightly smaller subset. This yields the quotient statement for flexible P‑stability.

Section 3 deals with stability in finite actions. The authors first use residual finiteness of (G) to embed an asymptotic homomorphism of (Q) into a genuine sofic approximation of (G) by taking a product with a finite quotient of (G). Then, assuming (G) is stable in finite actions, the limit action of this approximation is weakly contained in finite (G)‑actions. By a quantitative version of Kazhdan’s property (T) (Lemma 3.3), any almost‑invariant partition under the Kazhdan subgroup can be corrected to an exactly invariant one with error (O(\sqrt{\delta})). This correction allows the authors to descend the finite (G)‑action to a finite (Q)‑action that reproduces the statistics of the original approximation, establishing stability for (Q).

Section 4 introduces the notion of RFD for groups, i.e., the full group C*-algebra admits a separating family of finite‑dimensional *‑representations. The authors prove that if (N) has property (T) and (G) is RFD, then the quotient (Q) inherits the RFD property. The argument relies on the fact that Kazhdan’s property forces any almost‑invariant vector in a finite‑dimensional representation to be close to an invariant one, which in turn yields a genuine finite‑dimensional representation of the quotient separating any given non‑trivial element.

Section 5 discusses measured sofic approximations, a tool that allows one to treat the Bernoulli shift (G\curvearrowright{0,1}^Q) as a measurable model of the group. Section 6 shows that if this Bernoulli shift is sofic, then the quotient (Q) must be sofic as well; the proof again uses the Kazhdan subgroup to promote approximate invariance to exact invariance in the measurable setting.

Finally, Section 7 combines the abstract results with concrete constructions coming from variations of the Rips construction due to Wise and Belegradek–Osin. By carefully choosing a hyperbolic group (G) with a normal subgroup (N) that is either finitely generated or has property (T), the authors produce explicit counterexamples to several open questions:

• Corollary B: a hyperbolic, residually finite group that is not flexibly P‑stable.
• Corollary C: a hyperbolic, residually finite group that is not stable in finite actions.
• Corollary D: a hyperbolic group whose full group C*-algebra fails to be RFD.
• Corollary E: assuming the existence of a non‑sofic group, there exists a hyperbolic group admitting a non‑sofic p.m.p. action.

These examples are significant because prior to this work no hyperbolic group with such pathological approximation or stability behavior was known. The results illustrate that the presence of a Kazhdan normal subgroup can dramatically affect the approximation landscape of the ambient group and its quotients.

Overall, the paper provides a unified framework for understanding how approximation properties (soficity, flexible P‑stability, stability in finite actions) and representation‑theoretic finiteness (RFD) are transferred through extensions with finitely generated or Kazhdan normal subgroups. The combination of abstract group‑theoretic arguments with concrete Rips‑type constructions yields new examples that answer several longstanding open problems in geometric group theory, operator algebras, and measured dynamics.