I'm reading about uniform tree lattices. Let $X$ be a locally finite tree and $G$ its automorphism group. A subgroup $\Gamma\leq G$ that is discrete and such that $\Gamma\setminus X$ is finite is called a uniform tree lattice.
I'm having trouble coming up with a simple example. What would be the simplest (nontrivial) examples of a uniform tree lattice?
OK, let $F_n$ denote the free group on $n<\infty$ generators, $X$ the Cayley graph of $F_n$ with respect to the free generating set. Then $X$ is a locally finite tree (its valence equals $2n-1$), the group $F_n$ acts on $X$ freely and $X/F_n$ is a finite graph $R_n$, the wedge of $n$ circles. ($X$ is the universal cover of $R_n$.) Therefore, $F_n$ acts as a uniform tree lattice. Other interesting examples come from finite amalgams of finite groups, e.g. $$ Z_n \star Z_{2m} \star_{Z_m} Z_{2m}$$ etc. The book "Trees" by Serre is a classic treatment of group actions on trees. The book
"Tree lattices'', by H. Bass and A. Lubotzky, with appendices by H. Bass, L. Carbone, A. Lubotzky, G. Rosenberg, and J. Tits, Prog. Math., vol. 176, Birkhauser Boston, Inc., Boston, MA, 2001, xiv + 233 pp.
is another good reference.