I always think of Coxeter groups as defined by the Coxeter (or Coxeter-Dynkin) diagram. Then adding a new generator corresponds to extending the diagram by adding a node and various edges. So the answer to your question is simply "extending the diagram", I think. Neil On Tue, Jul 10, 2012 at 8:12 PM, Mike Stay <metaweta@gmail.com> wrote:
Is there a term for the operation of adding another generator to a Coxeter group? It has to be an involution, of course, but shouldn't satisfy any other relations---i.e. in the Coxeter matrix presentation, the off-diagonal elements of the new row and column should be infinity.
This turns up in the context of continued fractions. The infinite dihedral group acts on the real line translating by 2 and negating. If you do this operation, you get the (inf, inf, inf) hyperbolic triangle group, which acts on the upper half-plane in the same way plus geometric inversion, which is a hyperbolic reflection. If the three generators are s,t,u, then the continued fraction [a, b, c, ...] means the element ... u s^c u t^b u s^a.
It works abstractly on any Coxeter group, but since not all Coxeter groups have linear representations, it doesn't always have a nice geometric meaning. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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