Htykuut

Let $G$ be a finitely presented group and H a subgroup of index $n$ in $G$. Suppose that H has a non-trivial decomposition as amalgamated product, say $H = A ast_U B$. I am wondering about the following two questions:

(i) If $n = 2$, does then $G$ also have a non-trivial amalgamated decomposition?

(ii) More generally, does there exists sufficient conditions (for example on $A,B,U$) such that $G$ also have non-trivial amalgamated decomposition?

Yes, there are many examples: start from any group $A$, and consider $Awr C_2=A^2rtimes C_2$, $C_2$ permuting both copies. Consider the inclusion of index 2 $$A^2subset Awr C_2.$$

a) If $A$ has a nontrivial amalgam decomposition, so does the group $A^2$ (since it has $A$ as quotient group).

b) It remains the question when $Awr C_2$ has Property FA. This is answered in Theorem 1.1 of my paper with Aditi Kar (arxiv link, J. Group Theory publisher link). Assuming that $A$ is countable, and $F$ a nontrivial finite group, $Awr F=A^Frtimes F$ has Serre's Property FA if and only if $A$ is finitely generated and has finite abelianization. ($star$)

So, for (a) let's assume that $A$ has a nontrivial amalgam decomposition, and for (b) let's assume that $A$ is finitely generated with finite abelianization. In this case, $A^2$ admits a nontrivial amalgam decomposition, but not its overgroup of index two $Awr C_2$.

To satisfy both fulfillments, one can take $A$ infinite dihedral, or any Coxeter group with an amalgam decomposition (such as $langle a,b,c:a^2=b^2=c^2=(ab)^k=(bc)^m=1rangle$), or many other examples.

The same holds with the cyclic group $C_2$ replaced with $C_n$ for arbitrary $nge 2$.

To be self-contained here's a proof of ($star$). Let $A$ be a finitely generated group with finite abelianization, $F$ a nontrivial finite group, and let $G=Awr F$ act on a tree $T$ without edge inversion. Write $A^F=prod_{iin F}A_i$.

Suppose that the action is unbounded. Then it is unbounded on $A_i$ for some $i$, and hence on each $A_i$ since they are all conjugate.
Since $A$ is finitely generated, choose some element $f_i$ in $A_i$ acting as a loxodromic isometry; its axis $D_i$ is then invariant by $A_j$ for all $jneq i$. For $jneq i$, $D_j$ is $f_j$-invariant and hence $D_j=D_i$. This proves (using that $|F|ge 2$ to ensure the existence of $j$) that $D_i$ is $A_i$-invariant. Also it does not depend on $i$; call it $D$: $D$ is the unique minimal $A_i$-invariant subtree $T_i$ of $T$. Since $F$ permutes the $T_i$, it thus preserves the axis $D$.

Since the actions of each $A_i$ on the axis $D$ commute with each other and are all unbounded and $|F|ge 2$, they cannot be orientation reversing (since the centralizer of an orientation-reserving unbounded action on the line is trivial); so the action of $A_i$ on $D$ is given by a nontrivial homomorphism $A_itomathrm{Aut}^+(D)simeqmathbf{Z}$. But since $A$ has a finite abelianization, we deduce a contradiction.