### Coarse graph theory and applications to infinite groups

Combinatorics Seminar

27th October 2020, 11:00 am – 12:00 pm

Virtual (online) Zoom seminar; a link will be sent to the Bristol Combinatorics Seminar and Bristol Algebra & Geometry Seminar mailing lists, the week before the seminar.

A remarkably fruitful technique in modern infinite group theory is to represent a group using a graph and to determine relationships between the algebraic properties of the group and geometric and combinatorial properties of the graph. A single group can typically be represented by many graphs; this difficulty is typically overcome by studying properties of graphs which are "large-scale" (invariant under a family of maps called quasi-isometries). There are many of these.

A much less well studied problem is the following: given a group G and a subgroup H, the natural inclusion of a graph of H into a graph of G can be far from a quasi-isometry. We know of very few properties of graphs which are sufficiently robust to provide useful information in this context.

In my talk, I will discuss these problems, and introduce three coarse properties of graphs which provide helpful information on the subgroups of infinite groups.

*Biography:*

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