In particular, you must write the solution up on your own. Please acknowledge any cooperative work at the end of each assignment. Plagiarism, collusion, or other violations of the Academic Honor Principle , after consultation, will be referred to the The Committee on Standards. Please sign up for problems on the Homework cribsheet. Syllabus: [ PDF ] Syllabus This course will provide an introduction to the geometry of discrete groups.
The Geometry of Discrete Groups
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About us. Fact Sheet. Objective The research topic we propose lies in the intersection of Group Theory, Geometry and low-dimensional Topology. In this project we wish to explore the geometry and the topology at infinity of discrete groups. The geometrical viewpoint for groups has sparked the interest of geometers, topologists and group theorists since the seminal work of M. Gromov on the asymptotic invariants of groups.
We would like to look at groups from a topological viewpoint, and to study some topological properties at infinity of groups. In particular we will mainly focus on the geometric simple connectivity g.
The simple connectivity at infinity is an important tameness condition on the ends of the space, and it has been used to characterize Euclidean spaces among contractible open topological manifolds. Whereas the geometric simple connectivity is a related notion developed by V. It is worthy to note that it can be shown that all reasonable examples of groups e. Hence it would be very interesting to find an example of a finitely presented group which fails to be g.
Discrete groups which are not g.
Math 125: Geometry of Discrete Groups
The first step will be to find some combinatorial property equivalent to the g. On the other hand, if one can show that ANY group is geometrically simply connected, then the g. Both cases will have a deep impact in the understanding of the space of groups and for their geometrical classification.
Field of Science geometry topology. Activity type Higher or Secondary Education Establishments.
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Website Contact the organisation. Administrative Contact Nicolas Lecompte Mr. Status Closed project. Start date 1 September End date 31 August 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.
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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. Work performed and main results Among the most important asymptotic properties of a topological nature are the connectivity conditions at infinity i.
For a finitely generated group, the number of ends i. 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. 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.
- Final Report Summary - ASYMGTG (Asymptotic geometry and topology of discrete groups).
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