It is well known that [Burger and Mozes constructed][1] infinite, finitely presented, torsion-free simple groups which split as amalgams $F_n \ast_{F_{k}} F_m$ of finitely generated free groups, where $k$ is a certain function of $m$ and $n$. It is an interesting problem to make this amalgam as small as possible. For example, $F_{217} ∗_{F_{75601}} F_{217}$ is the smallest example constructed by Burger and Mozes; this was later improved to $F_9 \ast_{F_{81}} F_9$ by Rattaggi, and very recently to $F_5 \ast_{F_{25}} F_5$ by Caprace-Radu. These examples are crucially based on non-abelian simple groups, and in particular, the fact that $A_6$ is a non-abelian finite simple group such that the stabilizer of one letter in the action on $\{1, \dots, 6\}$ is isomorphic to another non-abelian finite simple group, namely $A_5$. For more details on this, see [Rattaggi's excellent compendium][2]. 

My question, then, concerns the following: is this the smallest (with respect to some ordering on the ranks of the factors and the amalgamated subgroup) possible example of such a Burger-Mozes amalgam? It is clearly (?) not doable using only the original techniques of Burger and Mozes, as $A_5$ is of course the smallest non-abelian finite simple group. 

I am also interested in any extensions of the original arguments to more general settings, e.g. by extending their normal subgroup theorem to a broader class of lattices. 

Note that the ordering chosen on the factors and the amalgamated subgroup may have some relevance. Furthermore, some amalgams are isomorphic; the original Burger-Mozes example, for example, satisfies $F_{217} ∗_{F_{75601}} F_{217} \cong F_{349} ∗_{F_{75865}} F_{349}$.


  [1]: http://www.numdam.org/article/PMIHES_2000__92__151_0.pdf
  [2]: http://www.rattaggi.ch/esq.pdf