on Cayley diagrams

Question

is the picture

the Cayley Graph of the group $\langle a,b,c\mid a^2, b^2,c^2\rangle$ ?

What would it be for $\langle a,b,c\mid a^2b^2c^2\rangle$?

Answer 1

Your Cayley graph seems to be correct for me. For $\langle a,b,c \mid a^2b^2c^2 \rangle$, it is not difficult to guess that its Cayley graph is an hexagonal tessalation of the plane. But to prove it you have to know how to solve the word problem of the presentation.

Hint: $\langle a,b,c \mid a^2b^2c^2 \rangle$ is the fundamental group of the connected sum of three projective planes $S_3= \mathbb{R}P^1 \# \mathbb{R}P^1 \# \mathbb{R}P^1$. It is know that $S_3$ can be constructed by pairwise identifying the opposite sides of an hexagon, so that $\pi_1(S_3)$ acts on the regular hexagonal tesselation of the plane with $S_3$ as quotient.