Burger and Mozes constructed (Burger and Mozes - Lattices in products of trees) infinite, finitely presented, torsion-free simple groups which split as amalgams of two finitely generated free groups over a finite index subgroup embedded by specific embeddings obtained through studying actions on product of regular trees. 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 finite 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.

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 $F_{217} ∗_{F_{75601}} F_{217} \cong F_{349} ∗_{F_{75865}} F_{349}$.

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    $\begingroup$ Could you make the question a little more precise? First, I want to emphasize that the notation $F_n\ast_{F_k}F_m$ is a bit misleading, because there are many ways to embed $F_k$ as a finite index subgroup of $F_n$ (as soon as there's one), and even specifiying the chosen finite index subgroups is not enough to specify the group: the subtlety of the constructions is precisely to choose these embeddings. Anyway, given such an an amalgam, one can associate a number, for instance $m+n$, or $m+n+\sqrt{k}$, and try to minimize it among possible simple such amalgams. $\endgroup$ – YCor Mar 1 '19 at 14:29
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    $\begingroup$ Side remark: in the 50's, the first amalgams of free groups that are simple were constructed; this was over infinitely generated subgroups and thus were infinitely presented groups. I think I once read (in a paper from the 50s too) that an amalgam of f.g. free groups over a finitely generated subgroup can be simple only if the amalgamated subgroup has finite index in both factors, although no example was known then (probably they were expected not to exist). $\endgroup$ – YCor Mar 1 '19 at 14:31
  • $\begingroup$ You're of course right on the point of using $F_k$; I wanted to emphasize the shape of the amalgam, rather than make a precise statement, but I see now that this can be misleading. I'll update it. The ordering you suggest I agree would make for a good order. $\endgroup$ – Carl-Fredrik Nyberg Brodda Mar 1 '19 at 14:34
  • $\begingroup$ Indeed, the first example you refer to is due to Camm in 1953. I am not aware of the second paper you refer to, but I'll try to see if I cannot find it. $\endgroup$ – Carl-Fredrik Nyberg Brodda Mar 1 '19 at 14:35

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