Geometric finiteness in large families in dimension 3

Geometric Diffeomorphism Finiteness in Low Dimensions and Homotopy Group Finiteness

Our main result asserts that for any given numbers C and D the class of simply connected closed smooth manifolds of dimension m < 7 which admit a Riemannian metric with sectional curvature bounded in absolute value by |K| ≤ C and diameter uniformly bounded from above by D contains only finitely many diffeomorphism types. Thus in these dimensions the lower positive bound on volume in Cheeger’s F...

The Geometric Finiteness Obstruction

The purpose of this paper is to develop a geometric approach to Wall's finiteness obstruction. We will do this for equivariant CW-complexes. The main advantage will be that we can derive all the formal properties of the equivariant finiteness obstruction easily from this geometric description. Namely, the obstruction property, homotopy invariance, the sum and product formulas, and the restricti...

Geometric Finiteness in Negatively Pinched Hadamard Manifolds

In this paper, we generalize Bonahon’s characterization of geometrically infinite torsion-free discrete subgroups of PSL(2,C) to geometrically infinite discrete torsionfree subgroups Γ of isometries of negatively pinched Hadamard manifolds X. We then generalize a theorem of Bishop to prove that every such geometrically infinite isometry subgroup Γ has a set of nonconical limit points with cardi...

Ascent of Finiteness of Flat Dimension

The main focus of this talk is to prove ascent of finiteness of flat dimension through local homomorphisms. Descent is known classically, e.g., it is in Cartan–Eilenberg’s book on homological algebra, but the corresponding ascent property is surprising because of the need to use derived categories. We present a short proof of ascent due to Dwyer–Greenlees–Iyengar, and discuss the main ingredien...

Geometric Sparsity in High Dimension