Acylindrical Hyperbolicity and Pure Conjugating Automorphisms in Graph Products of Groups

Abstract

We study graph products of groups over defining graphs of arbitrary cardinality from two closely related viewpoints. First, we characterize acylindrical hyperbolicity. If the defining graph is irreducible, has at least two vertices, and has a finite star base, then every parabolically full subgroup is either virtually cyclic or acylindrically hyperbolic. For vertex-full subgroups the finite star base condition is also necessary. In particular, the graph product itself is acylindrically hyperbolic exactly when the graph has a finite star base and the group is not virtually cyclic, or equivalently is not the infinite dihedral group. We also give the corresponding classification for reducible defining graphs. Second, motivated by our earlier work on countable right-angled Coxeter groups, we study pure conjugating automorphisms in the topology of pointwise convergence. We prove that they are topologically generated by finite-support factorwise automorphisms and partial conjugations, and establish closedness, density, exact-equality, and discreteness results. The same finite star base condition that appears in the acylindrical hyperbolicity criterion governs closedness and discreteness in the star-connected case; for arbitrary graphs, its natural refinement is the existence of a finite component witness set. Finally, for graph products of abelian groups we obtain a canonical topological semidirect decomposition of the pure conjugating automorphism group.