David Constantine

Marked length rigidity for one-dimensional spaces

In a compact geodesic metric space of topological dimension one, the minimal length of a loop in a free homotopy class is well-defined, and provides a function l : π1(X) → ℝ+ ∪{∞} (the value ∞ being assigned to loops which are not freely homotopic to any rectifiable loops). This function is the marked length spectrum. We introduce a subset Conv(X), which is the union of all non-constant minimal loops of finite length. We show that if X is a compact, non-contractible, geodesic space of topological dimension one, then X deformation retracts to Conv(X). Moreover, Conv(X) can be characterized as the minimal subset of X to which X deformation retracts. Let X1,X2 be a pair of compact, non-contractible, geodesic metric spaces of topological dimension one, and set Yi = Conv(Xi). We prove that any isomorphism ϕ : π1(X1) → π1(X2) satisfying l2 ∘ ϕ = l1, forces the existence of an isometry Φ : Y1 → Y2 which induces the map ϕ on the level of fundamental groups. Thus, for compact, non-contractible, geodesic spaces of ...

MARKED LENGTH RIGIDITY FOR FUCHSIAN BUILDINGS

DAVID CONSTANTINE AND JEAN-FRANC˘OIS LAFONTAbstract. We consider nite 2-complexes Xthat arise as quotients of Fuch-sian buildings by subgroups of the combinatorial automorphism group, whichwe assume act freely and cocompactly. Assume Xhas no vertex links whichare generalized 3-gons. We show that locally CAT(-1) metrics on Xwhich arepiecewise hyperbolic and satisfy a natural non-singularity condition at verticesare marked length spectrum rigid within the class of locally CAT(-1) piecewisenegatively curved metrics satisfying the same non-singularity condition.

On the Hausdorff Dimension of CAT(κ)Surfaces

Abstract We prove that a closed surface with a CAT(κ) metric has Hausdorff dimension = 2, and that there are uniform upper and lower bounds on the two-dimensional Hausdorff measure of small metric balls. We also discuss a connection between this uniformity condition and some results on the dynamics of the geodesic flow for such surfaces. Finally,we give a short proof of topological entropy rigidity for geodesic flow on certain CAT(−1) manifolds.