Final Report Summary - ASYMGTG (Asymptotic geometry and topology of discrete groups)
Project context and objectives
The general aim of the research project was to improve the knowledge on the behaviour of infinity in discrete groups. The study of groups as defined by their presentations is very old, and a general method to construct a group presentation for an arbitrary group is to see whenever it can act in a 'certain good way' on a 'nice' space (such as a manifold). To have a more satisfactory description of the relationship between such a space and any group which acts nicely on it, one should regard the group itself as a metric object, thus bringing geometry into the equation.
Our project was concerned more precisely with the study of the asymptotic topology and geometry of universal covers of compact spaces with a given fundamental group. The underlying idea was that all such possible topological models for a given group should share some robust geometric or topological properties (at infinity, also called asymptotic properties), the general goal being to explore and understand such 'global' topological and geometrical properties (for finitely generated groups).
To any finitely presented group one can associate some 'natural' spaces (the group itself with the word metric, Cayley graphs and Cayley 2 complexes) that depend upon the presentation but that are also 'similar' on a large scale (i.e. quasi-isometrics) and from this viewpoint the interesting properties of groups are those that are invariant under quasi-isometries (such a property is called 'geometric').
Work performed and main results
Among the most important asymptotic properties (of a topological nature) are the connectivity conditions at infinity (i.e. the condition that spheres that are 'very close' to infinity-bound balls, which are 'near' infinity).
For a finitely generated group, the number of ends (i.e. the number of connected components at the infinity of Cayley graphs) already gives (algebraic) information on its structure; in particular, it turns out that to understand the behaviour at infinity of finitely generated groups, we were led to study one-ended groups. With this in mind, for groups that are one-ended or simply connected at infinity (SCI), there is a very natural study that we initially formulated and researched within the project: the idea of measuring the 'minimal' way two points (resp. a loop) near infinity can be connected (resp. filled), and then to classify the different possible types of (simple) connectedness at infinity (trivial or not). More precisely, we defined a form of growth function for these conditions (the end-depth and the SCI growth) and we proved the linearity of the SCI growth for several classes of groups (Coxeter and Artin groups, lattices, amalgamated products), and, more interestingly, we proved that all one-ended groups have the same type of connectedness at infinity (i.e. they all have a linear end-depth).
One of the other notions we investigated during the project was Brick's quasi-simply filtered (QSF) property (where a space is said to be quasi-simply filtered if one can find an exhaustion of it that is 'approximable' by simply connected compacts). To better understand this notion, our first aim was to compare the QSF with other tameness conditions for groups and manifolds. We finally proved that the QSF property is a proper homotopy invariant and we were able to find several combinatorial condition equivalents for groups to the QSF. Thanks to these results, we increased our knowledge on these tameness conditions, their relationships and their combinatorial counterparts.
More recently, in collaboration with V. Poénaru, we obtained other results on the QSF property for groups, but from the low-dimensional topological viewpoint. Starting from our characterisation of QSF groups (which is that a finitely presented group is QSF if - and only if - there is a smooth compact manifold having it as a fundamental group and whose universal cover is geometrically simply connected), we used Poénaru's notion of 'representation' for finitely presented groups, where such a representation is roughly a map from some 2-complex to a certain singular 3-manifold associated to (the presentation of) a group, satisfying several topological properties. More precisely we defined the class of easily representable groups as the class of those finitely presented groups admitting a representation whose set of double points is closed. Our main result is that easily representable groups are QSF.
%Socio-economic impact of the project
Poénaru has developed a program for possibly showing that all groups are QSF. Our result can then be seen as a model for it and can serve as a comprehensible introduction. However, Poénaru generally works with non-easy representations, and so things become much more complicated and difficult to comprehend. Moreover, it is also worthy to note that, as for the QSF, there is no example of a group which does not admit an easy representation, so it is intriguing to ask if the QSF and the easy representability are equivalents for groups. With Poénaru we think that that answer is yes, and we are now working on this.
In conclusion, if the QSF property was really satisfied by all (finitely presented) groups, then it will be the very first example on a non-trivial geometric condition for groups that is valid for all of them, and this will 'stem' Gromov's philosophy that 'every property of all groups is either false or trivial'.
Skills acquired and career development
As a result of researching the project, the fellow vastly improved his expertise in the fields of geometric group theory and geometric topology, acquiring new knowledge and becoming familiar with new tools in CAT(0)-geometry, combinatorial group theory, group actions and low-dimensional topology and geometry. He broadened his knowledge in other fields of geometry and algebra, and also developed new ideas and research directions. He learnt how to work independently, how to develop contacts and how to find interesting questions to study.
The project has been a great opportunity to improve his skills and his personal career as a researcher: he strengthened his relationships with other universities and consolidated links and collaborations. Thanks to the knowledge and new skills acquired, he has his own field of study and has begun an independent career.
The general aim of the research project was to improve the knowledge on the behaviour of infinity in discrete groups. The study of groups as defined by their presentations is very old, and a general method to construct a group presentation for an arbitrary group is to see whenever it can act in a 'certain good way' on a 'nice' space (such as a manifold). To have a more satisfactory description of the relationship between such a space and any group which acts nicely on it, one should regard the group itself as a metric object, thus bringing geometry into the equation.
Our project was concerned more precisely with the study of the asymptotic topology and geometry of universal covers of compact spaces with a given fundamental group. The underlying idea was that all such possible topological models for a given group should share some robust geometric or topological properties (at infinity, also called asymptotic properties), the general goal being to explore and understand such 'global' topological and geometrical properties (for finitely generated groups).
To any finitely presented group one can associate some 'natural' spaces (the group itself with the word metric, Cayley graphs and Cayley 2 complexes) that depend upon the presentation but that are also 'similar' on a large scale (i.e. quasi-isometrics) and from this viewpoint the interesting properties of groups are those that are invariant under quasi-isometries (such a property is called 'geometric').
Work performed and main results
Among the most important asymptotic properties (of a topological nature) are the connectivity conditions at infinity (i.e. the condition that spheres that are 'very close' to infinity-bound balls, which are 'near' infinity).
For a finitely generated group, the number of ends (i.e. the number of connected components at the infinity of Cayley graphs) already gives (algebraic) information on its structure; in particular, it turns out that to understand the behaviour at infinity of finitely generated groups, we were led to study one-ended groups. With this in mind, for groups that are one-ended or simply connected at infinity (SCI), there is a very natural study that we initially formulated and researched within the project: the idea of measuring the 'minimal' way two points (resp. a loop) near infinity can be connected (resp. filled), and then to classify the different possible types of (simple) connectedness at infinity (trivial or not). More precisely, we defined a form of growth function for these conditions (the end-depth and the SCI growth) and we proved the linearity of the SCI growth for several classes of groups (Coxeter and Artin groups, lattices, amalgamated products), and, more interestingly, we proved that all one-ended groups have the same type of connectedness at infinity (i.e. they all have a linear end-depth).
One of the other notions we investigated during the project was Brick's quasi-simply filtered (QSF) property (where a space is said to be quasi-simply filtered if one can find an exhaustion of it that is 'approximable' by simply connected compacts). To better understand this notion, our first aim was to compare the QSF with other tameness conditions for groups and manifolds. We finally proved that the QSF property is a proper homotopy invariant and we were able to find several combinatorial condition equivalents for groups to the QSF. Thanks to these results, we increased our knowledge on these tameness conditions, their relationships and their combinatorial counterparts.
More recently, in collaboration with V. Poénaru, we obtained other results on the QSF property for groups, but from the low-dimensional topological viewpoint. Starting from our characterisation of QSF groups (which is that a finitely presented group is QSF if - and only if - there is a smooth compact manifold having it as a fundamental group and whose universal cover is geometrically simply connected), we used Poénaru's notion of 'representation' for finitely presented groups, where such a representation is roughly a map from some 2-complex to a certain singular 3-manifold associated to (the presentation of) a group, satisfying several topological properties. More precisely we defined the class of easily representable groups as the class of those finitely presented groups admitting a representation whose set of double points is closed. Our main result is that easily representable groups are QSF.
%Socio-economic impact of the project
Poénaru has developed a program for possibly showing that all groups are QSF. Our result can then be seen as a model for it and can serve as a comprehensible introduction. However, Poénaru generally works with non-easy representations, and so things become much more complicated and difficult to comprehend. Moreover, it is also worthy to note that, as for the QSF, there is no example of a group which does not admit an easy representation, so it is intriguing to ask if the QSF and the easy representability are equivalents for groups. With Poénaru we think that that answer is yes, and we are now working on this.
In conclusion, if the QSF property was really satisfied by all (finitely presented) groups, then it will be the very first example on a non-trivial geometric condition for groups that is valid for all of them, and this will 'stem' Gromov's philosophy that 'every property of all groups is either false or trivial'.
Skills acquired and career development
As a result of researching the project, the fellow vastly improved his expertise in the fields of geometric group theory and geometric topology, acquiring new knowledge and becoming familiar with new tools in CAT(0)-geometry, combinatorial group theory, group actions and low-dimensional topology and geometry. He broadened his knowledge in other fields of geometry and algebra, and also developed new ideas and research directions. He learnt how to work independently, how to develop contacts and how to find interesting questions to study.
The project has been a great opportunity to improve his skills and his personal career as a researcher: he strengthened his relationships with other universities and consolidated links and collaborations. Thanks to the knowledge and new skills acquired, he has his own field of study and has begun an independent career.