Some applications of -cohomology to the boundaries of Gromov hyperbolic spaces
Joint work with Bruce Kleiner.
Definition 1 Let be a contractible hyperbolic simplicial complex with bounded geometry. Its -cohomology is the following quotient space,
Theorem 2 (Strichartz, Bourdon-Pajot) If , then admits a measurable radial extension . Furthermore, the map mod constants descends to an injective map on cohomology.
Radial means that is the limit of along rays representing .
1.2. Numerical invariants
It is convenient to stick to continuous functions arising as radial extensions. These form an algebra of functions, from which further invariants can be extracted.
The equivalence relation on is defined by
Define numerical invariants
Theorem 4 (Kleiner, Carrasco) Assume is cocompact. Then
Theorem 5 (Bourdon-Pajot) Assume satisfies the combinatorial -Loewner property for some . Then
I will not define combinatorial Loewner property. It is quasi-Möbius invariant. It follows from the usual, analytic, Loewner property. The converse is conjectured to hold (Kleiner).
Example 1 Boundaries if rank one symmetric spaces, of Fuchsian buildings, are Loewner. Poincaré inequality implies Loewner.
2. New results
2.1. Ways to construct cohomology classes
The classical procedure is
Proposition 6 (Elek, Pansu) If , then .
I will use relative classes. Assume is a subcomplex. Then, by excision,
where denotes the boundary of as a subset of .
I will illustrate this construction in two situations.
2.2. Amalgamated products
Let , be hyperbolic groups, let be a quasi-convex and malnormal subgroup both of and of . Let . Take as (resp. ) the Cayley graph of (resp. ). Then is a disjoint union of copies of the Cayley graph of , indexed by .
Theorem 7 Assume is separable in , i.e. there exist a sequence of finite index subgroups of such that . Let . Then
- If , then for large enough,
- If , then, for and for large enough, the equivalence relation on is of the form
In other words, when replacing with , the cosets get far away from each other, and one can construct functions which are constant on these cosets, while keeping their gradients in .
Example 2 Let and have Sierpinski carpet boundaries with different conformal dimensions. Amalgamate them along peripheral Fuchsian subgroups. The result cannot have Loewner property.
2.3. Polygonal complexes
Theorem 8 Let be a simply connexted 2-complex such that
- Every 2-cell is isometric to an Euclidean polygon with at least sides.
- Every pair of 2-cells meet along at most a vertex or an edge.
- Every edge belongs to at most 2-cells.
(thus is hyperbolic). Then
In many cases, this upper bound is much better than previously known bounds (independent on ), based on area growth for the natural metric.
For this, one first constructs a tree in using geodesic segments in cells orthogonal to edges. Let be a thickening of that tree. The 2-cells in are polygons with two types of edges. It is not hard to compute when the relative cohomology of separates ends. Indeed, push frontier edges to infinity, replacing each cell by an ideal triangle. The result, like a hyperbolic building, admits retractions to hyperbolic plane. Pull back a smooth function on the closed disk. This yields a function which is constant on frontier edges, with controllable .
This construction seems specific to dimension 2.
Remark 1 increases under quasi-isometric embeddings, but does not.