Faithful Invariant Random Subgroups in Acylindrically Hyperbolic Groups

The paper proves Theorem 3.4 via Sun’s Cohen–Lyndon property and co-induction to obtain a weakly mixing, nontrivial, faithful IRS for every acylindrically hyperbolic group, with a complete and correct proof sketch (see the abstract and Theorem 3.4 together with Proposition 3.3 and its construction of µG via induction/co-induction, and the kernel/closure calculations) . For Theorem 3.6, the paper constructs an IRS inside Aut(G) supported in Inn(G) and then states that “we can see it as a characteristic random subgroup of G,” calling it faithful; this step is under-specified unless one assumes G is center-free, because passing from Inn(G) ≅ G/Z(G) back to Sub(G) by pullback forces the kernel to contain Z(G). The paper’s text gives no center-free hypothesis at that point . The candidate solution reproduces the core pipeline (Cohen–Lyndon + induction/co-induction) but uses a different base IRS (random selection of free factors) and explicitly handles the center issue, yielding a faithful characteristic IRS on G when Z(G)=1 and otherwise on G/Z(G), which fixes the missing hypothesis.