Here is one I like very much :

Let $F_s$ denote the free group on $s$ generators.

• If $F_s$ is a subgroup of finite index in $F_3$, then $s$ is odd.

More generally :

• assume we have an inclusion $F_s\subset F_t$ with $[F_t:F_s]=m$. Then $m=\frac{1-s}{1-t}$.

(Topological proof : we have a covering $F_t/F_s\to BF_s\to BF_t$ and we compute the Euler characterstics.)

Even more generally :

• Let $\Gamma$ be a torsion-free group of finite homological type (i.e. it has finite cohomological dimension, and a finite index subgroup of $\Gamma$ has finitely generated integral cohomology - any torsion-free arithmetic group satisfies these hypotheses). If $\Gamma'$ is torsion-free, then $m$ divides $\chi(\Gamma)$.

But the most spectacular in this vein seems to be

• If moreover $\Gamma$ is normal in $\Gamma'$, and $m$ is a prime power $p^r$ with $\text{gcd}(p,\chi(\Gamma))=1$, then $\Gamma\to\Gamma'\to Q$ is split !

This last theorem is due to K.S. Brown., and, applying it to $F_3$ we get : any extension $F_3\to\Gamma'\to Q$ with $Q$ a $p$-group of odd order is split.

Here is one I like very much :

Let $F_s$ denote the free group on $s$ generators.

• If $F_s$ is a subgroup of finite index in $F_3$, then $s$ is odd.

More generally :

• assume we have an inclusion $F_s\subset F_t$ with $[F_t:F_s]=m$. Then $m=\frac{1-s}{1-t}$.

(Topological proof : we have a covering $F_t/F_s\to BF_s\to BF_t$ and we compute the Euler characterstics.)

Even more generally :

• Let $\Gamma$ be a torsion-free group of finite homological type (i.e. it has finite cohomological dimension, and a finite index subgroup of $\Gamma$ has finitely generated integral cohomology - any torsion-free arithmetic group satisfies these hypotheses). If $\Gamma'$ is a torsion-free group containing $\Gamma$ as a subgroup of finite index $m$, then $m$ divides $\chi(\Gamma)$.

But the most spectacular in this vein seems to be

• If moreover $\Gamma$ is normal in $\Gamma'$, and $m$ is a prime power $p^r$ with $\text{gcd}(p,\chi(\Gamma))=1$, then $\Gamma\to\Gamma'\to Q$ is split !

This last theorem is due to K.S. Brown., and, applying it to $F_3$ we get : any extension $F_3\to\Gamma'\to Q$ with $Q$ a $p$-group of odd order is split.

Here is one I like very much :

Let $F_s$ denote the free group on $s$ generators.

• If $F_s$ is a subgroup of finite index in $F_3$, then $s$ is odd.

More generally :

• assume we have an inclusion $F_s\subset F_t$ with $[F_t:F_s]=m$. Then $m=\frac{1-s}{1-t}$.

(Topological proof : we have a covering $F_t/F_s\to BF_s\to BF_t$ and we compute the Euler characterstics.)

Even more generally :

• Let $\Gamma$ be a torsion-free group of finite homological type (i.e. it has finite cohomological dimension, and a finite index subgroup of $\Gamma$ has finitely generated integral cohomology - any torsion-free arithmetic group satisfies these hypotheses). If $\Gamma'$ is torsion-free, then $m$ divides $\chi(\Gamma)$.

But the most spectacular in this vein seems to be

• If moreover $\Gamma$ is normal in $\Gamma'$, and $m$ is a prime power $p^r$ with $\text{gcd}(p,\chi(\Gamma))=1$, then $\Gamma\to\Gamma'\to Q$ is split !

This last theorem is due to K.S. Brown., and it applying it to $F_3$ we get : any extension $F_3\to\Gamma'\to Q$ with $Q$ a $p$-group of odd order is split.

Here is one I like very much :

Let $F_s$ denote the free group on $s$ generators.

• If $F_s$ is a subgroup of finite index in $F_3$, then $s$ is odd.

More generally :

• assume we have an inclusion $F_s\subset F_t$ with $[F_t:F_s]=m$. Then $m=\frac{1-s}{1-t}$.

(Topological proof : we have a covering $F_t/F_s\to BF_s\to BF_t$ and we compute the Euler characterstics.)

Even more generally :

• Let $\Gamma$ be a torsion-free group of finite homological type (i.e. it has finite cohomological dimension, and a finite index subgroup of $\Gamma$ has finitely generated integral cohomology - any torsion-free arithmetic group satisfies these hypotheses). If $\Gamma'$ is torsion-free, then $m$ divides $\chi(\Gamma)$.

But the most spectacular in this vein seems to be

• If moreover $\Gamma$ is normal in $\Gamma'$, and $m$ is a prime power $p^r$ with $\text{gcd}(p,\chi(\Gamma))=1$, then $\Gamma\to\Gamma'\to Q$ is split !

This last theorem is due to K.S. Brown., and, applying it to $F_3$ we get : any extension $F_3\to\Gamma'\to Q$ with $Q$ a $p$-group of odd order is split.