= Solution
Suppose $w\ne1$ in $P$. In $D=P*F(z,d)$, the <commutator> $c=wzw^{-1}z^{-1}$ is a nonempty cyclically alternating free-product word. Every positive power remains reduced, so $c$ has infinite order. Each $g_j$ also has infinite order: this is immediate for $z,d$; if $x_i=1$, then $x_i z=z$, and otherwise $x_i z$ is a cyclically alternating word of length two.
Consequently the presentation defining $H$ is a multiple <HNN extension> identifying the infinite cyclic subgroups $\langle c\rangle$ and $\langle g_j\rangle$. By the <normal form theorem for an HNN extension>, $D$ embeds in $H$.
The subgroup
$$
U=\langle C,u_0,\ldots,u_{n+1}\rangle\le H
$$
is a <free group> on exactly these $n+3$ elements. For a proof, view $D$ as $(P*\langle z\rangle)*\langle d\rangle$. For every $k\ne0$, the reduced expression $C^k=d^{-1}c^kd$ lies neither in $P*\langle z\rangle$ nor in $\langle d\rangle$. It therefore belongs to none of the associated cyclic subgroups $\langle c\rangle$ or $\langle g_j\rangle$. Any freely reduced word in $C,u_0,\ldots,u_{n+1}$ with a stable letter has no pinch and is nontrivial by <Britton's lemma>. A word with no stable letters is a nonzero power of $C$, also nontrivial. This proves the claimed <free basis of a group>.
Inside the rank-two <free group> $\langle a_1,a_3\rangle\le Q$, the elements
$$
a_1,\ a_3a_1a_3^{-1},\ldots,\ a_3^{n+2}a_1a_3^{-(n+2)}
$$
freely generate a subgroup $V$ of rank $n+3$. This is the same free-product normal-form calculation as for the conjugate basis in Question 5: between conjugates with distinct indices, a nonzero power of $a_3$ remains. The identifications in the output presentation give an <isomorphism> $U\to V$ matching these free bases. Hence
$$
P(w)\cong H*_{U=V}Q.
$$
The <normal form theorem for an amalgamated free product> makes both factors embed. Composing the embeddings gives
$$
\boxed{P\hookrightarrow D\hookrightarrow H\hookrightarrow P(w).}
$$
Thus the construction preserves $P$ whenever the input word is nontrivial, regardless of whether that word has finite or infinite order in $P$.
Back to article page
