Subgroups and Combinatorial Nonpositive Curvature

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The proposed research circulates around the principal investigator's deep interest in cubical geometry and its applications. Highlights of the proposal include the following: 1) A genuinely new proof of the Malnormal Special Quotient Theorem. 2) An explicit characterization of the class of groups that are virtually limit groups. 3) A cubulation result for full relatively hyperbolic amalgams.3) A new and more general form of nonpositively curvature for square complexes. 4) A concrete and explicit virtual cubulation theorem for 3-manifolds. 5) A proposal to prove that irreducibilty is generic for cocompact lattices in the products of trees. 6) An investigation of virtual algebraic fibering for special cube complexes beyond that obtained in the 3-manifold setting. 7) A continued search for a unified theory of coherence. 9) An exploration of growth-gaps.