1. End of the proof of Theorem 10
Last week, we proved the first half of the following theorem.
Theorem 1 Let be a hyperbolic group. There exist constants , such that
- Every admissible graph has rank .
- For every admissible graph such that rank, there are at most admissible graphs with rank .
Now we prove the second half.
Proof: Let have rank . Denote by the set of admissible graphs with rank .
We prove that there exists exactly one extremal function such that .
Recall that is the set of conex with . Also, . We show that there is an injective map to the set of onempty subsets of . For , there is a unique extremal function such that . Then splits into even and odd components, and even components are bipartite, , and .
Define be the set of pointed cones where and is an edge of . We claim that the whole graph can be reconstructed from the set .
Indeed, if and is an edge of both and , then , so . Conversely, suppose that for all -edges . Among these, there is an edge of . Since , there is which is an edge of , and . The vertex of these pointed cones coincide. By the -point Lemma, and are connected by a path of length in , so .
One inclusion is easy since . Conversely, if and , then it follows that , so .
Clearly, the partition determines and hence .
Remark 1 This gives an exponential bound on the dimension of the injective hull.
We do not know how sharp it can be.
2. Proof of main result for hyperbolic groups
Theorem 2 (Lang, Moezzi 2010) Let be a hyperbolic group. Then is proper and equal to . The collection , admissible graph, is a locally finite polyhedral structure on with finitely many isometry types of cells, and acts properly and cocompactly on by cellular isometries.
Proof: We have already shown that is finite dimensional and locally finite. Since is discretely geodesic, has only finitely many isometry types of cells. This shows that is complete, and so agrees with .
Injective hull is unique up to isometries preserving and uniquely determined by their restriction to this subspace, so this gives the -action on . Since is a neighborhood of , acts cocompactly on .
3. Fixed point theorems
3.1. The case of non expansive maps
Theorem 3 (Sine, Soardi 1979) Let be an injective metric space. Let be -Lipschitz. Then
- If is bounded, then .
- If , then is injective and hence contractible.
Question: If is unbounded but has bounded orbits, is ? Answer turns out to be no.
Example 1 (Prus) Let , let . Then is isometric, it has no fixed points, but orbits are bounded.
Note that Prus’s map is not onto.
3.2. The case of isometries
Theorem 4 (Lang 2011) Let be injective. Let be a group of isometries of .
- If has bounded orbits, then .
- If , then is injective and hence contractible.
Proof: First we show that if is a metric space, a group of isometries with bounded orbits, then there is that is constant on each orbit.
Take two orbits , . Define
This has the properties of a metric except that unless is a fixed point. Let be the set of all functions such that
By Zorn’s Lemma, the poset has a minimal element . Consider as a function which is constant on orbits, . By construction,
so . By minimality of , for all and there exists such that
so , hence .
If is injective, , so for some . constant on orbits implies that along , so is a single point, fixed by .
Assuming that , we show hyperconvexity. Let and be such that . Since is hyperconvex, is non empty, and it is hyperconvex again (by definition…). For all , and ,
so . Similarly, so . Hence has a fixed point in , so , which we wanted.
3.3. Structure of fixed point sets in , hyperbolic
Since the cells of are affine, barycentric subdivision makes sense: For an admissible graph , define the center as the barycenter of vertices of . Let be the collection of simplices corresponding to strictly ascending sequences
still acts by simplicial isometries on . This is a “-CW-complex”, i.e. if an element of maps a simplex of to itself, then it fixes the simplex point wise.
Theorem 5 is a finite model for the classifying space for proper actions.
There are few examples of infinite groups where I can compute . Mainly, groups that act on trees.
Example 2 with obvious generating set.
The Cayley graph is a tree where vertices are blown up alternately into pentagons or squares. The injective hull fills in the holes with -squares, it is -dimensional, whereas Rips’ complex is -dimensional.
In fact, in general, cut points of give cut points in .
Example 3 with generating set .
Then is a diagonal strip in with group elements forming a staircase.
Example 4 .
The Cayley graph is a tree of alternating triangles and squares. I suspect that (not yet checked), combinatorially, the injective hull is a -dimensional cube complex.
I can prove that injective hulls of hyperbolic groups, when they are -dimensional, can be remetrized to become . Call flag a pair where and . There are types of -dimensional flags, correponding to vertical/horizontal (type 1) or diagonal (type 2) sides of cells.
Proposition 6 Let , , be admissible graphs, such that is a flag of type then is also admissible, and is a flag of opposite type .
Links are made of sectors of with angles a multiple of . The figure formed by such sectors contains a tripod but no midpoints, so it cannot be injective. Therefore the link condition for is satisfied.
Sisto: Is it true that maximal cells have the same dimension ? Answer: This may depend on the generating set. Look at . There are generating systems which yield -metric. It seems to me that in general, there is a notion of -modelled word metric.
Silberman: Your examples have a lot of torsion. What is the injective hull of a torsion free group like ?