Sending Loxodromic Elements of Right-Angled Artin Groups to Pseudo-Anosov Mapping Classes: A Partial Answer to Problem 2.28 of the K3 List

Problem 2.28 of the list K3: A New Problem List in Low-Dimensional Topology (Baykur, Kirby and Ruberman, eds., 2026), recorded by T. Koberda, asks: if \(\Gamma\) is a finite graph that is not a join, is there an injective homomorphism from the right-angled Artin group \(A(\Gamma)\) to the mapping class group of a surface which sends every loxodromic element to a pseudo-Anosov mapping class? We give a partial answer; the problem remains open in general. We prove that such a homomorphism exists, with a closed surface as target, when \(\Gamma\) is chordal, when the complement graph of \(\Gamma\) is chordal, when \(\Gamma\) is the complement of a cycle of length at least five (this includes the pentagon), and more generally when the complement of \(\Gamma\) is obtained from complete graphs and cycles by gluing along cliques. The answer is also positive for disjoint unions of graphs of these kinds and of joins, with isolated vertices added, and for connected graphs whose right-angled Artin groups have finite index in the group of a connected graph with a positive answer. Further constructions treat the hexagon, graphs obtained by duplicating vertices, and complements of cycles with one more vertex. Together with a computer enumeration of small graphs, the constructions give a positive answer for every graph with at most six vertices that is not a join and for \(758\) of the \(853\) such graphs with seven vertices; four more of these are settled by the finite-index property, and for the remaining \(91\) we have no answer. The homomorphisms are those of Clay, Leininger and Mangahas: the generators are sent to pseudo-Anosov maps of subsurfaces, and by a theorem of these authors the property asked for is equivalent to a filling condition, which we verify. The proofs use subsurface projections (Masur–Minsky, Behrstock) and, for complements of cycles and for the hexagon, planar configurations of discs lifted to totally ramified cyclic covers of the sphere. This is an unrefereed note.