Abstract:We prove that a lacunary hyperbolic group $G = \varinjlim G_i$ with sufficient generics is selfless in the sense of Amrutam--Gao--Kunnawalkam Elayavalli--Patchell, provided the hyperbolicity constants $\delta_i$ and injectivity radii $r_i$ satisfy $\delta_i(\log r_i)^{7} = o(r_i)$. The proof replaces the acylindricity-based machinery of that work with a direct geodesic $n$-gon criterion due to Arzhantseva, which applies in any $\delta$-hyperbolic space. As a consequence, combined with rapid decay, $G$ is $C^*$-selfless. The condition is mild: torsion-free Tarski monsters, Jacobson's mixed-identity-free elementary amenable groups and Gromov monster groups satisfy it for appropriate parameter choices. The amenable examples are selfless but cannot be $C^*$-selfless, providing examples that separate these properties. Finally we remark that the Gromov monster group examples provide a potential avenue to a non-exact $C^*$-algebra with strict comparison.
From: Martin Finn-Sell [view email]
[v1]
Thu, 18 Jun 2026 16:39:29 UTC (14 KB)
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