Non-inner amenability of the Higman–Thompson groups

We prove that the Higman–Thompson groups [Formula: see text] and [Formula: see text] are non-inner amenable for all [Formula: see text]. This extends Haagerup and Olesen’s result that Thompson’s groups [Formula: see text] and [Formula: see text] are non-inner amenable. Their proof relied on machinery only available in the [Formula: see text] case, namely, Thurston’s piecewise-projective model for Thompson’s group T, so, our approach necessarily utilizes different tools. This also provides an alternate proof of Haagerup–Olesen’s result when [Formula: see text].