Final Activity Report Summary - GEOMSTRUCT3MFDS (Geometric structures on 3-manifolds)
The research project 'Geometric structure on 3-manifoldS' fits in the framework of low-dimensional topology and geometry. The world as we see it has a naive notion of dimension. A line is one-dimensional, a surface is a two-dimensional object and solid objects are three-dimensional, but also the space-part of the universe is three-dimensional. Topology studies the shape of objects, up to continuous transformations.
Geometry gives a more accurate description of objects. There are many kinds of basic geometries, for example Euclidean geometry, but it is well-known that this is not the only one. William P. Thurston, in his pioneering work in the 1970s, showed that in dimension three there are only eight basic geometries, and it turns out that the most common is the hyperbolic one, where, of course, Euclid's axiom is not satisfied.
The main part of this project was devoted to the study of three-dimensional hyperbolic manifolds. One way to do so was to use a hyperbolic LEGO. More precisely, for studying hyperbolic manifolds we defined basic hyperbolic pieces and then glued them together, as in a LEGO game, to build our objects. Formally speaking, this was called the theory of geodesic ideal triangulations. In the project we studied, classified and generalised geodesic ideal triangulations of hyperbolic manifolds.
A set of instructions, formally its fundamental group, was associated to each hyperbolic manifold, telling us how to walk in the hyperbolic space. In the project, we studied representations of fundamental groups of hyperbolic three-manifolds and the main outcome we obtained was a volume-rigidity result. The volume covered in a walk which followed the instructions of the representation was always smaller than the hyperbolic volume of the corresponding manifold, and it was maximal only when such instructions came from a hyperbolic LEGO game. This result, proved in his full generality by Francaviglia and Klaff, needed the work of many mathematicians, including Thurston, Gromov, Goldman and Dunfield, which begun 20 years ago.
The third research line that was followed was the study of free groups. This is a very important field of modern mathematics. For instance, each time one sweeps its credit card, groups are at work. The theory of groups, which has in fact applications in algorithmics and combinatorics, has many branches, one of them being the so-called geometric group theory. The general principle was to study a group via its geometric representations.
In terms of the obtained results, firstly, Francaviglia proved a length compactness theorem, stating that there were only finitely many representations of a free group that had bounded length, meaning that the length of a transformation was somehow the average of the lengths of the paths we were walking in. This was an important open problem and had as a corollary the solution of a conjecture of Ilya Kapovich in theory of algorithms. It also had another nice application. In theory of algorithms, it is very important to study the speed of algorithms, since the method for encoding passwords has to be very fast, while we must be sure that the methods for cracking codes are very, very slow. The Whitehead algorithm has the strange property that it is in theory very slow, but in fact very fast, and the study of lengths on free groups gives a theoretic explanation of this fact.
For a given free group, the set of one-dimensional objects having the group as fundamental is called outer space. The same definition can be given for surfaces, and in this case we find the so-called Teichmüller space. Teichmüller space and outer space are very similar, but the latter is not as well-understood as the first.
In the project we studied outer space, and our main result, in collaboration with Armando Martino, was to successfully find a natural metric, i.e. a way to measure distances on it, so that geometric tools could now be used for studying it.
Geometry gives a more accurate description of objects. There are many kinds of basic geometries, for example Euclidean geometry, but it is well-known that this is not the only one. William P. Thurston, in his pioneering work in the 1970s, showed that in dimension three there are only eight basic geometries, and it turns out that the most common is the hyperbolic one, where, of course, Euclid's axiom is not satisfied.
The main part of this project was devoted to the study of three-dimensional hyperbolic manifolds. One way to do so was to use a hyperbolic LEGO. More precisely, for studying hyperbolic manifolds we defined basic hyperbolic pieces and then glued them together, as in a LEGO game, to build our objects. Formally speaking, this was called the theory of geodesic ideal triangulations. In the project we studied, classified and generalised geodesic ideal triangulations of hyperbolic manifolds.
A set of instructions, formally its fundamental group, was associated to each hyperbolic manifold, telling us how to walk in the hyperbolic space. In the project, we studied representations of fundamental groups of hyperbolic three-manifolds and the main outcome we obtained was a volume-rigidity result. The volume covered in a walk which followed the instructions of the representation was always smaller than the hyperbolic volume of the corresponding manifold, and it was maximal only when such instructions came from a hyperbolic LEGO game. This result, proved in his full generality by Francaviglia and Klaff, needed the work of many mathematicians, including Thurston, Gromov, Goldman and Dunfield, which begun 20 years ago.
The third research line that was followed was the study of free groups. This is a very important field of modern mathematics. For instance, each time one sweeps its credit card, groups are at work. The theory of groups, which has in fact applications in algorithmics and combinatorics, has many branches, one of them being the so-called geometric group theory. The general principle was to study a group via its geometric representations.
In terms of the obtained results, firstly, Francaviglia proved a length compactness theorem, stating that there were only finitely many representations of a free group that had bounded length, meaning that the length of a transformation was somehow the average of the lengths of the paths we were walking in. This was an important open problem and had as a corollary the solution of a conjecture of Ilya Kapovich in theory of algorithms. It also had another nice application. In theory of algorithms, it is very important to study the speed of algorithms, since the method for encoding passwords has to be very fast, while we must be sure that the methods for cracking codes are very, very slow. The Whitehead algorithm has the strange property that it is in theory very slow, but in fact very fast, and the study of lengths on free groups gives a theoretic explanation of this fact.
For a given free group, the set of one-dimensional objects having the group as fundamental is called outer space. The same definition can be given for surfaces, and in this case we find the so-called Teichmüller space. Teichmüller space and outer space are very similar, but the latter is not as well-understood as the first.
In the project we studied outer space, and our main result, in collaboration with Armando Martino, was to successfully find a natural metric, i.e. a way to measure distances on it, so that geometric tools could now be used for studying it.