Algorithms for subgroups and automorphisms of right-angled Artin Groups

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A finitely-presented group is a right-angled Artin group (RAAG) if the only relations between the generators are that some pairs of generators commute. The P.I. will study the subgroup structure and automorphism groups of RAAGs. The main goal of this project is to develop algorithms for analyzing RAAGs; these algorithms will generalize classical algorithms including Nielsen reduction, Stallings folding, column reduction, and Whitehead's algorithm. Goal applications include an algorithm for testing membership in automorphism group orbits in RAAGs, and restricted versions of the subgroup membership problem. A second goal is to define combinatorial and topological objects on which automorphism groups of RAAGs act, and to use these actions to better understand the automorphism groups. A third goal is to prove homological finiteness results about important subgroups of automorphism groups of RAAGs; this part of the project will focus on automorphism groups of free groups specifically.A right-angled Artin group (RAAG) is a type of algebraic structure in which all equations are consequences of equations asserting that certain pairs of elements commute (equations of the form "x*y=y*x"). The P.I. will research the structure of general RAAGs, specifically the substructures of these objects, and their symmetries (automorphisms). RAAGs and their substructures and symmetries are important objects of study in geometric group theory; these groups include special cases that have been studied since the origins of group theory. Free abelian groups (for example the lattice of integer points in coordinate n-space) are examples of RAAGs; their automorphism groups are matrix groups, and many questions about their structure and symmetry can be answered using familiar matrix techniques such as row and column reduction. A class of groups called free groups are also examples of RAAGs. Nielsen reduction and Whitehead's algorithm are combinatorial algorithms from the 1920's and 1930's for analyzing certain aspects of free groups, and Stallings folding is a related graph-theoretical technique from the 1970's. The main goal of the project is to construct common generalizations of these classical free group algorithms and matrix techniques that would simultaneously apply to all RAAGs, or to prove that no such generalizations exist.