3 Rewriting end to use Stallings paper.

Suppose that we are given a PL map from a handlebody $W$ to a handlebody $V$. Choose a spine for $W$. Homotope the map until the image is a regular neighborhood of the image of the spine. By general position, our map is now an embedding. Fix a pants decomposition of disks $D = (D_i)$, for $V$. Suppose that $P$ is a component of $V - D$ (so $P$ is a three-ball with three distinguished disks on its boundary). Consider a component $X$ of $f(W) \cap P$. This is essentially a knotted graph. Via a homotopy (keeping $X \cap D$ fixed) unknot $X$. If the rank of $X$ is positive then another homotopy produces small lollipops which we shall compress a bit later. If $X$ meets any disk $D_i \subset \partial P$ more than once then we may homotope a leg of $X$ through $D_i$. Let $Y$ be the resulting component of $f(W) \cap P$. (Note that this reduces $f(W) \cap D$.)

Homotoping in this fashion we eventually arrive at an embedding of $W$ so that every component in every three-ball of $V - D$ is either a tripod or an interval, possibly with lollipops attached. The feet of the tripod/interval lie in distinct disks in the boundary of the containing solid pants.

Now compress all of the lollipops to get $f'(W')$ (a new handlebody, because we compressed and a new map because we have to extend it over the two-handles we added).

I claim that $f'$ is $\pi_1$ injective and further $W'$ embeds into a finite cover of V, using subgroup separability.

However

EDIT: reading what I just wrote I believe that I have essentially reproduced a small This reproduces, in our context, part of Stallings paper (eg sliding the leg is a fold, arriving at only tripods and intervals is producing produces an immersion...immersion.)
So I'll stop here.

Since $f'$ is an immersion, it follows from Stallings paper that $f'$ is $\pi_1$ injective and that $W'$ embeds into a finite cover of $V$.

2 fix typo

Suppose that we are given a PL map from a handlebody $W$ to a handlebody $V$. Choose a spine for $W$. Homotope the map until the image is a regular neighborhood of the image of the spine. By general position, our map is now an embedding. Fix a pants decomposition of disks $D = (D_i)$, for $V$. Suppose that $P$ is a component of $V - D$ (so $P$ is a three-ball with three distinguished disks on its boundary). Consider a component $X$ of $f(W) \cap P$. This is essentially a knotted graph. Via a homotopy (keeping $X \cap D$ fixed) unknot $X$. If the rank of $X$ is positive then another homotopy produces small lollipops which we shall compress a bit later. If $X$ meets any disk $D_i \subset \partial P$ more than once then we may homotope a leg of $X$ through $D_i$. Let $Y$ be the resulting component of $f(W) \cap P$. (Note that this reduces $f(W) \cap D$.)

Homotoping in this fashion we eventually arrive at an embedding of $W$ so that every component in every three-ball of $V - D$ is either a tripod or an interval, possibly with lollipops attached. The feet of the tripod/interval lie in distinct disks in the boundary of the containing solid pants.

Now compress all of the lollipops to get $f'(W')$ (a new handlebody, because we compressed and a new map because we have to extend it over the two-handles we added).

I claim that $f'$ is $\pi_1$ injective and further $W'$ embeds into a finite cover of V, using subgroup separability.

However: reading what I just wrote I believe that I have essentially reproduced a small part of Stallings paper (eg sliding the leg is a fold, arriving at only tripods and intervals is producing an immersion...)
So I'll stop.stop here.

1

Suppose that we are given a PL map from a handlebody $W$ to a handlebody $V$. Choose a spine for $W$. Homotope the map until the image is a regular neighborhood of the image of the spine. By general position, our map is now an embedding. Fix a pants decomposition of disks $D = (D_i)$, for $V$. Suppose that $P$ is a component of $V - D$ (so $P$ is a three-ball with three distinguished disks on its boundary). Consider a component $X$ of $f(W) \cap P$. This is essentially a knotted graph. Via a homotopy (keeping $X \cap D$ fixed) unknot $X$. If the rank of $X$ is positive then another homotopy produces small lollipops which we shall compress a bit later. If $X$ meets any disk $D_i \subset \partial P$ more than once then we may homotope a leg of $X$ through $D_i$. Let $Y$ be the resulting component of $f(W) \cap P$. (Note that this reduces $f(W) \cap D$.)

Homotoping in this fashion we eventually arrive at an embedding of $W$ so that every component in every three-ball of $V - D$ is either a tripod or an interval, possibly with lollipops attached. The feet of the tripod/interval lie in distinct disks in the boundary of the containing solid pants.

Now compress all of the lollipops to get $f'(W')$ (a new handlebody, because we compressed and a new map because we have to extend it over the two-handles we added).

I claim that $f'$ is $\pi_1$ injective and further $W'$ embeds into a finite cover of V, using subgroup separability.

However: reading what I just wrote I believe that I have essentially reproduced a small part of Stallings paper (eg sliding the leg is a fold, arriving at only tripods and intervals is producing an immersion...)
So I'll stop..