First thing about GGT I want to remark is very positive: it’s straightforward; one doesn’t need to read heavy books with definitions to acquaint with actual problems. After this first day I have a feeling that GGT (or rather GGT represented by people on this specific school/conference) is very close to what Lieven Le Bruyn described here as a healthy branch of mathematics.
Whole day was full of lectures: apart from coffe breakes there was only one long (2 hours) break for a lunch. First lecture was at 9:00 (which is way to early for me, I tried to sleep on almost every coffe break) and last question session was at 19:20. Actually, it’s a bit to much for me, as some lecturers are handing out some exercises and there is virtually no time to solve them. But to much is better then to little :-).
There is parallel conference taking place here on the topic of differential equations, so there’s some additional fun from talking with all these guys who know how to solve PDEs and stuff. God I wish I had this easiness with deltas and epsilons. :-)
Here are some notes from today lectures.
Two main courses at the conference are “Simplicial Non-Positive Curvature” and “Cat(0) cubical complexes”. These are supplemented with one-hour talks about some new results.
(*Simplicial Non-Positive Curvature*)
We’ll be interested in simplicial complexes. People often try to express some properties from other branches of geometry – like differential geometry – in combinatorial terms that would be applicable to general simplicial complexes. This is because simplicial complexes are easier to understand then, say, manifolds – after all a simplicial complex can be easily presented as some very finite combinatoric data. For example, there is a Gauss-Bonnet theorem for 2-dimensional simplicial complex X:
Σ (3-α(v)) + Σ (6-α(v)) = 6∙χ(X),
where first sum is over vertices v lying on the boundary of X, the second sum is over vertices lying in the interior of X, and α(v) is numberof triangles meeting in v (and χ(X) is Euler characteristic). This is, more or less, consequence of Euler’s formula. So as you see by comparison with classical Gauss-Bonet, the number of triangles meeting in a vertex say how “negatively curved” the space is in this vertex. This is quite natural when compared to imaginable examples of non-positively curved manifolds – I always imagined “negatively curved points” on such manifolds as being in a saddle (hence “having more space” around then points have on Euclidean plane)
(So simplicial complexes can be thought as a limit of Riemannian manifolds in which the curvature is “focused in single points”)
Troughout the lecture we used this Gauss-Bonnet theorem only in case X=2-disk. Then the proof of this identity is for sure just a consequence of Euler’s formula – by for example takin two copies of this disk and gluing them into a 2-sphere.
As far as I understand, one of the founding fathers of Geometric Group Theory is Mikhail Gromov. Some years ago he posed more or less the following problem: Find a combinatorial condition for a simplicial complex that would guarantee that this complex is non-positively curved in a metric sense.
Metric space is said to be non-positively curved iff it is a geodesic (that is, between any two points there exist a geodesic, and geodesic is in this context by definition a locally shortest path) space and all triangles are “thick”.
The last condition means that if we map isometrically a triangle (triangle = three points plus geodesic segments that connect them)
onto a triangle in Euclidean plane then any two points in this Euclidean triangle are nearer to each other on Euclidean plane then their counterimages are in the space under consideration.
Non positively curved metric space is also called a Hadamard space and a CAT(0) space – 0 is because we compared triangles with triangles on a Euclidean plane; if we compared them with triangles on a sphere we would get a CAT(1) space, and if we compared them with triangles on hyperbolic plane we’d get a CAT(-1) space.
A theorem by Cartan and Alexandroff says that Riemannian manifolds with non-positive sectional curvature are in fact non-positively curved spaces. Who’d thought that :-).
Simplicial Non-Positive Curvature is not really an answer to a Gromov question, but it was motivated by it. It’s a rather simple combinatorial condition for simplicial complexes. Curiously, it has quite many implications similar to CAT(0) for metric spaces (like, say, asphericality (this links to wikipedia article coauthored by me :-)). For this reason some believe (this was part of some other talk) that there should some underlying meta-theory that would contain both CAT(0) and SNPC spaces.
So let’s define this SNPC condition. Simplicial complex is flag iff everytime it contains a set of points joined pairwisely by 1-simplices it also contains a simplex spanned by this set of points.
Simplicial complex is k-large, k≥4, iff it is flag and every cycle γ in X of lenght 3<|γ|<k has a diagonal (So for k=4 we get just flag and for k=5 we get “no empty square condition”)
Simplicial complex is k-systolic iff it is connected, simply connected and k-large.
And finally, 6-systolicity (which is often shortened to “sistolicity”) is precisely SNPC condition. Weird as it may seem at first sight, it has some interesting properties ;-).
First, I gave perhaps the shortest of possible definitions. Now I’ll give a longer one, but one which is almost purely local – so one can perhaps see better why it might be similar in some sense to CAT(0) condition.
For this, few more definitions: for a given simplex σ of simplicial complex X we define a link X_σ to be “a sphere of radius 1 around σ”. That is X_σ is a subcomplex of X consisting of all those simplices that are disjoint with σ but which span a simplex in X with σ.
Systol sys(X) of a simplicial complex X is a lenght of a shortest cycle that doesn’t have a diagonal (for example, “minimal” homotopically non-trivial cycles don’t have diagonals, but also a boundary of clasically triangulated hexagon doesn’t have a diagonal).
Now, the theorem says that a simplicial complex X is k-systolic iff sys(X)≥k (this is “nonlocal”) and sys(X_σ)≥k for every simplex σ (this is “local”). It;s not very difficult to prove.
It’s 1AM here, so I’ll end eith giving some (non)examples of systolic complexes:
- torus with quite a tricky triangulation coming from a hexagon with identifications on boundary
- Triangulation of Euclidean plane by equilateral triangles
- Triangulation of Hyperbolic plane by equilateral traingles with angle 2π/6
- Some triangulation of Hyperbolic 3-space (there was a pictureon a blackboard but I didn’t get it
- Cartesian product of a tree and a line has a systolic triangulation, but a product of two nontrivial tree hasn’t.
- It follows easily from above Gauss-Bonnet theorem that a sphere doesn’t have a systolic triangulation (because G-B says that there exist a vertex adjacent to less than 6 triangles, so the link of v is a circle of lenght < 6 …). As a consequence, manifolds of dimension ≥3 are not systolic (because in manifolds links of codim 3 simplices are 2-spheres)
Well, unfortunately I didn’t get very far, but writing this took me ~2 hours. Tomorrow I’ll change somehow the mode of writing.