Geometric Group Theory and the Topology of Aspherical Manifolds

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AbstractAward: DMS-0104026Principal Investigator: Michael W. DavisThis is a proposal for research in geometric group theoryfocusing on Coxeter groups, Artin groups, and mapping classgroups of surfaces. The main problems to be addressed are thefollowing.(1) For which Coxeter groups is the Coxeter diagram uniquelydetermined by the group? For which Coxeter groups is thefundamental generating set uniquely determined (up toconjugation) by the group? For which Coxeter groups is the outerautomorphism group finite?(2) Are all Artin groups linear groups? (3) Find a formula for the cohomology with compact supports of abuilding. Likewise for the Salvetti complex of an Artin group.(4) Determine the l^2 Betti numbers of cubical manifoldsassociated to right-angled Coxeter groups. Do they vanish outsidethe middle dimension? This would imply the Flag ComplexConjecture concerning triangulations of odd-dimensional spheresand has implications for graph embeddings.(5) Develop a theory of mock reflection groups. This is a class ofgroups similar to Coxeter groups which arise as transformationsof blow-ups of hyperplane arrangements.(6) Is the Torelli subgroup of the mapping class groups of asurface of genus at least three finitely generated?(7) Can a word hyperbolic group be the fundamental group of asurface by surface bundle? Must all finitely presented non wordhyperbolic groups contain a Baumslag-Solitar group or an abeliangroup of rank two?Group theory arises from the study of symmetries of anobject. When this object has an interesting geometric structure,one can use geometric techniques to better understand the groupof symmetries. This project involves the study of certainfamilies of groups which arise in a broad range of mathematicaland physical contexts, such as the study of crystal structuresand the intertwining of DNA. These groups are associated to richand beautiful geometric structures which lend themselves to thetechniques of geometric group theory.