Cayley Graphs and Cakes

Over the past month I have been studying at the AMSI Summer School at La Trobe University in Melbourne. Eight courses are offered at the AMSI Summer School and I took the one on geometric group theory. Geometric group theory is also the subject of my honours thesis and a great area of mathematics. I previously wrote about some ideas from geometric group theory here.

One of the main ideas in geometric group theory is to take a finitely generated group and turn it into a geometric object by constructing a Cayley graph (or in Germany, a Dehn gruppenbild or Dehn group picture).

If G is a group with a generating set A, then the Cayley graph of (G,A) has the elements of the group as vertices and for each group element g \in G and generator a \in A, there is an edge from g to ga.

Cayley graphs can be very pretty and geometrical interesting. In the final week of the course, our homework was to creatively make a Cayley graph. Here’s a sample of the Cayley graphs we made.

With a friend I baked a cake and decorated it with the Cayley graph of the group C_2 * C_3 \cong \langle a ,b \mid a^2,b^3 \rangle with respect to the generating set \{a,b\}. We were really happy with how it looked and tasted and are proud to say that the whole thing got eaten at a BBQ for the summer school students.

Staying with the food theme, a friend use grapes and skewers to make their Cayley graph. It’s a graph of the discrete Heisenberg group \langle a,b,c \mid ac=ca, bc=cb, ba=abc \rangle. I was amazed at the structural integrity of the grapes. There’s a video about this Cayley graph that you can watch here (alternatively it’s the first result if you seach “Heisenberg group grapes”).

This Cayley graph is made out of paper and shows how picking a different generating set A can change the appearance of the Cayley graph. It is a Cayley graph of \mathbb{Z}^2. Normally this Cayley graph is drawn with respect to the generating set \{(1,0),(0,1)\} and Cayley graph looks like the square lattice on the right. The Cayley graph on the left was made with respect to the generating set \{(1,0),(0,1),(1,1)\} and the result is a triangular tiling of the plane. Note that while the two Cayley graphs of \mathbb{Z}^2 look quite different, they have some important shared properties. In particular, both of them are two dimensional and flat. (The second image is taken from here)

Someone also made a Cayley graph of the Baumslag Solitar group \text{BS}(1,2) \cong \langle a,t \mid tat^{-1}=a^2 \rangle with respect to \{a,t\}. This was a group that came up frequently in the course as it can be used to construct a number of surprising counter examples.

Finally, my favourite Cayley graph of the day was the incredibly pretty Cayley graph of the Coxeter group

\langle a,b,c,d \mid a^2, b^2, c^2, d^2, (ab)^3, (bc)^3, (ac)^2, (ad)^2, (cd)^2 \rangle

with respect to the set \{a,b,c,d\}.

I’d like to thank both AMSI and La Trobe for putting on the Summer School and the three geometric group theory lecturers Alejandra Garrido, Murray Elder and Lawrence Reeves for teaching a great course. A huge thanks to the other students for making some great Cayley graphs and letting me share some of them.

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