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The question is simple:

For a $3$-manifold $M$, if $\pi_1(M)$ contains a surface group $\Gamma$ (i.e. the fundamental group of some surface) then $M$ contains a $\pi_1$-injective immersed surface $S$.

Existence of $S$ obviously ensures the existence of $\Gamma$ but I don't know if the converse is true. Actually the statement I found is from here which says that

Showing that every hyperbolic $3$-manifold contains an immersed $\pi_1$-injective surface is equivalent to showing that the fundamental group of every hyperbolic $3$-maniofld contains a subgroup isomorphic to the fundamental group of a surface.

Is this true because the assumption $M$ here is hyperbolic? (because it's $K(G,1)$ space?)

Edit: I didn't add a condition that $\pi_1(S) = \Gamma$ by mistake. The actual question is the existence of a $\pi_1$-injective immersed surface with $\pi_1(S) = \Gamma$.

N.B. Although I accepted the answer because it answers the originally stated question, it's still not clear the existence of an $\pi_1$-injective immersed surface in $M$ with $\Gamma$ as the fundamental group.

The question is simple:

For a $3$-manifold $M$, if $\pi_1(M)$ contains a surface group $\Gamma$ (i.e. the fundamental group of some surface) then $M$ contains a $\pi_1$-injective immersed surface $S$.

Existence of $S$ obviously ensures the existence of $\Gamma$ but I don't know if the converse is true. Actually the statement I found is from here which says that

Showing that every hyperbolic $3$-manifold contains an immersed $\pi_1$-injective surface is equivalent to showing that the fundamental group of every hyperbolic $3$-maniofld contains a subgroup isomorphic to the fundamental group of a surface.

Is this true because the assumption $M$ here is hyperbolic? (because it's $K(G,1)$ space?)

Edit: I didn't add a condition that $\pi_1(S) = \Gamma$ by mistake. The actual question is the existence of a $\pi_1$-injective immersed surface with $\pi_1(S) = \Gamma$.

The question is simple:

For a $3$-manifold $M$, if $\pi_1(M)$ contains a surface group $\Gamma$ (i.e. the fundamental group of some surface) then $M$ contains a $\pi_1$-injective immersed surface $S$.

Existence of $S$ obviously ensures the existence of $\Gamma$ but I don't know if the converse is true. Actually the statement I found is from here which says that

Showing that every hyperbolic $3$-manifold contains an immersed $\pi_1$-injective surface is equivalent to showing that the fundamental group of every hyperbolic $3$-maniofld contains a subgroup isomorphic to the fundamental group of a surface.

Is this true because the assumption $M$ here is hyperbolic? (because it's $K(G,1)$ space?)

Edit: I didn't add a condition that $\pi_1(S) = \Gamma$ by mistake. The actual question is the existence of a $\pi_1$-injective immersed surface with $\pi_1(S) = \Gamma$.

The question is simple:

For a $3$-manifold $M$, if $\pi_1(M)$ contains a surface group $\Gamma$ (i.e. the fundamental group of some surface) then $M$ contains a $\pi_1$-injective immersed surface $S$.

Existence of $S$ obviously ensures the existence of $\Gamma$ but I don't know if the converse is true. Actually the statement I found is from here which says that

Showing that every hyperbolic $3$-manifold contains an immersed $\pi_1$-injective surface is equivalent to showing that the fundamental group of every hyperbolic $3$-maniofld contains a subgroup isomorphic to the fundamental group of a surface.

Is this true because the assumption $M$ here is hyperbolic? (because it's $K(G,1)$ space?)

Edit: I didn't add a condition that $\pi_1(S) = \Gamma$ by mistake. The actual question is the existence of a $\pi_1$-injective immersed surface with $\pi_1(S) = \Gamma$.

N.B. Although I accepted the answer because it answers the originally stated question, it's still not clear the existence of an $\pi_1$-injective immersed surface in $M$ with $\Gamma$ as the fundamental group.

The question is simple:

For a $3$-manifold $M$, if $\pi_1(M)$ contains a surface group $\Gamma$ (i.e. the fundamental group of some surface) then $M$ contains a $\pi_1$-injective immersed surface $S$.

Existence of $S$ obviously ensures the existence of $\Gamma$ but I don't know if the converse is true. Actually the statement I found is from here which says that

Showing that every hyperbolic $3$-manifold contains an immersed $\pi_1$-injective surface is equivalent to showing that the fundamental group of every hyperbolic $3$-maniofld contains a subgroup isomorphic to the fundamental group of a surface.

Is this true because the assumption $M$ here is hyperbolic? (because it's $K(G,1)$ space?)

The question is simple:

For a $3$-manifold $M$, if $\pi_1(M)$ contains a surface group $\Gamma$ (i.e. the fundamental group of some surface) then $M$ contains a $\pi_1$-injective immersed surface $S$.

Existence of $S$ obviously ensures the existence of $\Gamma$ but I don't know if the converse is true. Actually the statement I found is from here which says that

Showing that every hyperbolic $3$-manifold contains an immersed $\pi_1$-injective surface is equivalent to showing that the fundamental group of every hyperbolic $3$-maniofld contains a subgroup isomorphic to the fundamental group of a surface.

Is this true because the assumption $M$ here is hyperbolic? (because it's $K(G,1)$ space?)

Edit: I didn't add a condition that $\pi_1(S) = \Gamma$ by mistake. The actual question is the existence of a $\pi_1$-injective immersed surface with $\pi_1(S) = \Gamma$.