Consider the matrices $A = \frac{1}{5}\begin{pmatrix}5&0&0\\\ 2&2&1\\\ 2&1&2\end{pmatrix}$, $B = \frac{1}{5}\begin{pmatrix}2&2&1\\\ 0&5&0\\\ 1&2&2\end{pmatrix}$, and $C = \frac{1}{5}\begin{pmatrix}2&1&2\\\ 1&2&2\\\ 0&0&5\end{pmatrix}$.

The group $G\subset GL_3(\mathbb{C})$ they generate is free of rank 3. However, one also has:

$A^\infty := \lim_{k\rightarrow\infty}A^k = \begin{pmatrix}1&0&0\\\ 1&0&0\\\ 1&0&0\\\ \end{pmatrix}$, and similarly $B^\infty = \begin{pmatrix}0&1&0\\\ 0&1&0\\\ 0&1&0\\\ \end{pmatrix}$ and $C^\infty = \begin{pmatrix}0&0&1\\\ 0&0&1\\\ 0&0&1\\\ \end{pmatrix}$.

This leads to three "infinite" relations among $A$,$B$, and $C:$

$A^\infty B = B^\infty A$

$B^\infty C = C^\infty B$

$C^\infty A = A^\infty C$

Of course these are not true group relations since they involve infinite products and furthermore these infinite products are singular, hence do not actually lie in the group generated by $A$, $B$, and $C$, but rather are relations among elements of the "boundary" of this group (in an appropriate sense of the word boundary).

Question 1:I am wondering if anyone has studied such infinite/boundary relations on groups, and in particular if they have any usefulness in understanding any properties of the original group.

Question 2:Suppose $H$ is another free group of rank 3 in $GL_3(\mathbb{C})$. $H$ is of course isomorphic to $G$, but may not have any boundary relations. Is there a way to use the infinite relations to define a "stronger" notion of isomorphism which says that $G$ and $H$ are not isomorphic since they do not have the same infinite relations?Edit:Upon further thought, this latter question is more of a question about the particular representation of $G$ chosen, so perhaps it is not so interesting? Or maybe something aside from the usual stuff about isomorphism of representations might be relevant?

**Motivation:** Awhile back I was studying harmonic functions on the Sierpinski Gasket; these three matrices arise naturally in the evaluation of such functions. In this context the infinite relations above are equivalent to the statement that such functions are well-defined at each point of the Gasket. The relations above were occasionally useful for proving various facts, but at the time I never bothered to consider them in the context of group relations.

More generally they can arise in studying subgroups of $GL_n(R)$ for $R$ a commutative unital ring. In this case the "boundary" of the subgroup may be a subset of the singular elements of $Mat_n(R)$ and one can ask questions about relations between elements of this "boundary".

Gromovboundary. That gives you a keyword to search for, by the way. – Mariano Suárez-Alvarez♦ Mar 15 '11 at 17:59Metric spaces of non-positive curvature. – HJRW Mar 16 '11 at 3:35