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.