= Free subgroups of every finite rank in the free product of Z and C2
{c}
Let $G=\langle x,y\mid y^2=1\rangle$. For each $n\geq2$, a free basis of a subgroup of rank $n$ is
$$
\{x^{n-1}\}\ \cup\ \{x^j yxyx^{-j}:0\leq j\leq n-2\}.
$$
A covering-space proof uses $r=n-1$ spheres double-covering the <real projective plane>, with antipodal points $a_j,b_j$ over the wedge point. Add lifted $x$ edges in a cyclic order through the $a_j$, and an $x$ loop at each $b_j$. This is a connected $2r$-sheeted cover of a circle wedged with a projective plane. After collapsing one connecting arc in each simply connected sphere, its fundamental group is that of a graph with $r$ vertices and $2r$ edges, hence free of rank $r+1$.
A spanning tree along the first $r-1$ cyclic edges gives the displayed words: the closing cycle is $x^r$, and the loop reached through $a_j,b_j$ is $x^j yxyx^{-j}$. Injectivity of the induced fundamental-group map of a <covering map> proves they form a free basis in $G$.
Back to article page
