Deck transformation group

The deck transformation group consists of all deck transformations under composition. For a connected regular covering corresponding to a normal subgroup $H◃{\pi }_1(X)$, it is isomorphic to ${\pi }_1(X)/H$.

If a finite group $G$ is generated by $h$ elements, send the $a$-generators of the genus-$h$ surface group to those generators and its $b$-generators to the identity. The surface relation is satisfied, so this is a surjective group homomorphism. Its kernel defines a connected regular covering space, on which the deck transformation group $G$ acts freely. The lifted orientation and multiplicativity of the Euler characteristic give genus $g=1+|G|(h-1)$.

Every finitely generated group acts freely and properly discontinuously by Riemannian isometries on a connected manifold in any dimension $d\ge 3$. Choose a surjection $F_r\to T$ and the regular covering of ${\#}_r(S^1\times S^{d-1})$ associated with its kernel. Pulling back a metric makes the deck transformation group isometric. Uniqueness of lifts makes its action free. The cover can be noncompact and need not be simply connected; no claim about realizing every group as a closed three-manifold fundamental group is involved.

Let $p:E\to X$ be a connected covering space, with $X$ path-connected and locally path-connected, and let $F=p^{-1}(x_0)$. Restriction to $F$ identifies the deck transformation group with the centralizer of the monodromy action of a covering space inside the symmetric group $\mathrm{Sym}(F)$.

A deck transformation commutes with lifting every based loop, so its restriction centralizes the monodromy action of a covering space. Conversely, let $\sigma$ be a permutation of $F$ that centralizes this action. Represent $e\in E$ as the endpoint of a lifted path $\gamma$ starting at $a\in F$, and send $e$ to the endpoint of the lift of $\gamma$ starting at $\sigma (a)$. Two such representations differ by a lifted based loop; commutation with its monodromy proves independence of the representation.

On a path-connected evenly covered neighborhood, this construction carries each sheet to a sheet by the corresponding local homeomorphism. It is therefore continuous. Applying the construction to ${\sigma }^{-1}$ gives its inverse, hence a deck transformation. Since $E$ is path-connected, a deck transformation is determined by one point, proving uniqueness and the asserted group isomorphism.