# Groups with presentation $\langle x_1,x_2,\dotsc, x_n\mid x_1^3, x_2^3,\dotsc, x_n^3\rangle$

I'm computer engineer but I'm working in some topics related with group theory. I found (accidentally) a group with presentation

$\langle x_1,x_2,\dotsc, x_n\mid x_1^3, x_2^3,\dotsc, x_n^3\rangle$

with some interesting topological properties (from the point of view of graph theory and automatic structures). My question is does this presentation belong to some family of groups? What happen if we change the exponent of relator?. What about the growth function of this group?

Thanks very much.

• Maybe knowing the presentation could help us
– Francesco Polizzi
Jan 30, 2014 at 15:41
• My apologies. I don't know why it doesn't appear. The presentation is <x_1, x_2, ... , x_ n|(x_1)^3, (x_2)^3, ..., (x^n)^3>. thanks.
– Miguel C.
Jan 30, 2014 at 15:48
• This is $C_3 * C_3 * \ldots *C_3$, i.e. the free product of $n$ copies of the cyclic group of order $3$. Look at en.wikipedia.org/wiki/Free_product
– Francesco Polizzi
Jan 30, 2014 at 15:59

The (spherical) growth function is $(1+2x)/(1-2(n-1)x)$.