Existence of a C*-selfless but non-C*-exact finitely generated group

### What is the conjecture

There exists a finitely generated group whose reduced $C^\*$-algebra is $C^\*$-selfless but not $C^\*$-exact.

**Context:** A $C^\*$-algebra $A$ is called $C^\*$-selfless if it satisfies a quantitative strengthening of the mixed-identity-free (MIF) property, introduced by Amrutam–Gao–Kunnawalkam Elayavalli–Patchell. The conjecture asks whether this strong algebraic regularity property can coexist with failure of $C^\*$-exactness. The paper identifies Gromov monster groups (constructed via geometric small cancellation over expander graphs) as candidate examples: they are known to be non- $C^\*$-exact, and the paper shows they are selfless under appropriate parameter choices. What remains is to establish the rapid decay property for these groups, which would promote selflessness to $C^\*$-selflessness.

If true, this would provide the first known example of a non-exact $C^\*$-algebra with strict comparison.

* **Source Paper:** *Lacunary hyperbolic groups with fast injectivity radius growth and enough loxodromic elements are selfless* ([arXiv:2606.20456v1](https://arxiv.org/abs/2606.20456v1), [HTML](https://arxiv.org/html/2606.20456v1), [PDF](https://arxiv.org/pdf/2606.20456v1))
* **Authors:** Goulnara Arzhantseva, Martin Finn-Sell
* **In-Paper Location:** [Conjecture 5.4 (Page 9)](https://arxiv.org/pdf/2606.20456v1#page=9) in Section 5 "Non-acylindrically hyperbolic examples of groups that are selfless"
* **OpenConjecture ID:** 3548 ([OpenConjecture](https://github.com/davisrbr/conjectures-arxiv))

### Prerequisites needed

* **Finitely generated groups:** [`Mathlib.GroupTheory.FreeGroup.Basic`](https://leanprover-community.github.io/mathlib4_docs/Mathlib/GroupTheory/FreeGroup/Basic.html), [`Mathlib.Algebra.Group.Subgroup.Basic`](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Group/Subgroup/Basic.html)
* **C\*-algebras:** [`Mathlib.Analysis.CStarAlgebra.Basic`](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/CStarAlgebra/Basic.html)
* **Not in Mathlib:** Reduced group C\*-algebra $C_r^\*(G)$; $C^\*$-exactness; the selfless / $C^\*$-selfless property (Amrutam–Gao–Kunnawalkam Elayavalli–Patchell); rapid decay property; Gromov monster groups

### Choose either option
- [ ] I plan on adding this conjecture to the repository
- [x] This issue is up for grabs: I would like to see this conjecture added by somebody else

This issue was generated by an AI agent and reviewed by me.