**Gromov Boundary**

This simple folklore proof uses dynamical properties of the action of hyperbolic elements on the Gromov boundary of a Gromov-hyperbolic group. For the special case of the free group *F*_{n}, the boundary (or space of ends) can be identified with the space *X* of semi-infinite reduced words

*g*_{1}*g*_{2}···

in the generators and their inverses. It gives a natural compactification of the tree, given by the Cayley graph with respect to the generators. A sequence of semi-infinite words converges to another such word provided that the initial segments agree after a certain stage, so that *X* is compact (and metrizable). The free group acts by left multiplication on the semi-infinite words. Moreover any element *g* in *F*_{n} has exactly two fixed points *g*±∞, namely the reduced infinite words given by the limits of *g**n* as *n* tends to ±∞. Furthermore *g**n*·*w* tends to *g*±∞ as *n* tends to ±∞ for any semi-infinite word *w*; and more generally if *w*_{n} tends to *w*≠ *g* ±∞, then *g**n*·*w*_{n} tends to *g*+∞ as *n* tends to ∞.

If *F*_{n} were boundedly generated, it could be written as a product of cyclic groups *C*_{i} generated by elements *h*_{i}. Let *X*_{0} be the countable subset given by the finitely many *F*_{n}-orbits of the fixed points *h*_{i} ±∞, the fixed points of the *h*_{i} and all their conjugates. Since *X* is uncountable, there is an element of *g* with fixed points outside *X*_{0} and a point *w* outside *X*_{0} different from these fixed points. Then for some subsequence (*g*_{m}) of (*g*n)

*g*_{m}=*h*_{1}*n*(*m*,1) ···*h*_{k}*n*(*m*,*k*), with each*n*(*m*,*i*) constant or strictly monotone.

On the one hand, by successive use of the rules for computing limits of the form *h**n*·*w*_{n}, the limit of the right hand side applied to *x* is necessarily a fixed point of one of the conjugates of the *h*_{i}'s. On the other hand, this limit also must be *g*+∞, which is not one of these points, a contradiction.

Read more about this topic: Boundedly Generated Group, Free Groups Are Not Boundedly Generated

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### Famous quotes containing the word boundary:

“No man has a right to fix the *boundary* of the march of a nation; no man has a right to say to his country, “Thus far shalt thou go and no further.””

—Charles Stewart Parnell (1846–1891)