Suppose we have a finite graph with vertex set V and edge set E. One can associate a group presentation to this graph as follows: each v in V corresponds to a generator of the group and we insist v has order 2 in the group. If v and w are connected by an edge in the graph, then we insist that the corresponding generators (or order 2) commute. The resulting group is a right-angled Coxeter group. Suppose V has n elements. If E is empty, then the resulting group is a free product of n copies of the group of order 2. If the defining graph is complete, then the resulting group is the direct product of n copies of the group of order 2. Thus the class of right-angled Coxeter groups interpolates between these two extreme cases.
In this talk, we will discuss the automorphism group of such a group. In particular, we would like to understand how the structure of the defining graph is related to the algebraic structure of the automorphism group of the right-angled Coxeter group. If the graph is complete, then the right-angled Coxeter group is finite and then so is its automorphism group. If there are no edges in the graph, then the automorphism group is infinite and quite interesting. We have a graph-theoretic condition that guarantees the automorphism group of the right-angled Coxeter group is not terribly complicated in the sense that there are only finitely many outer automorphisms. In this case, we can also prove that the resulting automorphism group is virtually a CAT(0) group.
Very little is known about the Geometric Group Theory of these groups. For example, it is unknown whether the no-edge case automorphism groups are CAT(0) or not. On the one hand, these groups can be realized as subgroups of an appropriate automorphism group of a non-abelian free group which may lead you to believe they don't have a chance of being CAT(0). On the other hand, they are closely related to the n-strand braid groups where the CAT(0) question is still open for 6 or more strands.
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