MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## “Brunnian” type groups? [closed]

I am not sure about a suitable name for the type of group that I am going to discuss and in the title I have used the word "Brunnian", which was borrowed from "Brunnian links" in topology. Recall that a Brunnian link is a nontrivial link that becomes trivial if any component is removed.

Now I am considering a type of groups with similar properties: A group $G$ of $n$ gengrators $x_1,\cdots,x_n$ such that in $G$, for any fixed $1\leq i\leq n$, if any element $y$ is can be expressed as a product of $x_1,\cdots,\hat{x_i},\cdots,x_n$ (that is, there is an expression of $y$ which contains no powers of $x_i$), then $y$ is equal to the identity element in $G$.

I just wonder if there is any known results for this type of groups. Has anyone done research with these groups before? A particular question that I am interested is to find out the cardinality of the center of a "Brunnian type" group, or at least give an upper bound of the order of the $p$-torsion component of the center.

Any comments and references will be greatly appreciated.

-
In your groups $x_i=1$ for every $i$? If so, the group is trivial (if $n>1$). – Mark Sapir May 4 2012 at 23:39
A meaningful question would be where (for every $i$) all generators but $x_i$ generate a free subgroup (like in the case of 1-relator groups). – Misha May 5 2012 at 4:19