Notice Board

DAVE MORRIS
University of Lethbridge

Topic: Colour-permuting automorphisms of complete Cayley graphs

Monday, March 17, 2025
12:15 - 1:15 p.m.
Markin Hall M1060

Abstract:  A bijection f of a metric space is "distance-permuting" if the distance from f(x) to f(y) depends only on the distance from x to y. For example, it is known that every distance-permuting bijection of the Euclidean plane is the composition of an isometry and a dilation (x --> kx). So they are affine maps. We study the analogue in which G is any (finite or infinite) group, and the "distance" from x to y is the "absolute value" of the unique element s of G, such that xs = y. We determine precisely which groups have the property that every distance-preserving bijection is an affine map. The smallest exception is the quaternion group of order 8, and all other exceptions are constructed from this one. It is natural to state the problem in the language of graph-theory: construct a graph by joining each pair of points (x,y) with an edge, and label (or "colour") this edge with its length. Then we are interested in bijections that permute the colours of the edges: i.e., the colour of the edge from f(x) to f(y) depends only on the colour of the edge from x to y. This is joint work with Shirin Alimirzaei.