Deck Transformations and Group Actions

For a covering space \(p:\tilde{X} \rightarrow X\) the isomorphisms \(\tilde{X} \rightarrow \tilde{X}\) are called deck transformations or covering transformations. These form a group \(G(\tilde{X})\) under composition. For example, for the covering space \(p: \mathbb{R} \rightarrow S^1\) projecting a vertical helix onto a circle, the deck transformations are the vertical translations taking the helix onto itself, so \(G(\tilde{X}) \approx \mathbb{Z}\) in this case. For the \(n\)-sheeted covering space \(S^1 \rightarrow S^1,\, z \mapsto z^n\), the deck transformations are the rotations of \(S^1\) through angles that are multiples of \(2\pi / n\), so \(G(\tilde{X})=\mathbb{Z}_n\).

By the unique lifting property, a deck transformation is completely determined by where it sends a single point, assuming \(\tilde{X}\) is path-connected. In particular, only theidentity deck transformation can fix a point of \(\tilde{X}\).

A covering space \(p:\tilde{X} \rightarrow X\) is called normal if for each \(x \in X\) and each pair of lifts \(\tilde{x},\tilde{x}'\) of \(x\) there is a deck transformation taking \(\tilde{x}`to :math:\)tilde{x}’. For example, the covering space :math:mathbb{R} rightarrow X^1` and the \(n\)-sheeted covering spaces \(S^1 \rightarrow S^1\) are normal. Intuitively, a normal covering space is one with maximal symmetry. This can be seen in the covering spaces of \(S^1 \vee S^1\) shown in the talbe earlier in this section, where the normal covering spaces are (1),(2),(5)-(8), and (11). Note that in (7) the group of deck transformations is \(\mathbb{Z}_4\) while in (8) it is \(\mathbb{Z}_2 \times \mathbb{Z}_2\).

Sometimes normal covering spaces are called regular covering spaces. The term ‘normal’ is motivated by the following result.

Proposition 1.39. Let \(p:(\tilde{X}, \tilde{x}_0) \rightarrow (X, x_0)\) be a path-connected covering space of the path-connected, locally path-connected space \(X\), and let \(H\) be the subgroup \(p_*(\pi_1(\tilde{X},\tilde{x}_0)) \subset \pi_1(X,x_0)\). Then:

1. This covering space is normal iff \(H\) is a normal subgroup of \(\pi_1(X,x_0)\).

2. \(G(\tilde{X})\) is isomorphic to the quotient \(N(H)/H\) where \(N(H)\) is the normalizer of \(H\) in \(\pi_1(X,x_0)\).

In particular, \(G(\tilde{X})\) is isomorphic to \(\pi_1(X,x_0)/H\) if \(\tilde{X}\) is a normal covering. Hence for the universal cover \(\tilde{X} \rightarrow X\) we have \(G(\tilde{X}) \approx \pi_1(X)\).

Proof: We observed earlier in the proof of the classification theorem that changing the basepoint \(\tilde{x}_0 \in p^{-1}(x_0)\) to \(\tilde{x}_1 \in p^{-1}(x_0)\) corresponds precisely to conjugating \(H\) by an element \([\gamma] \in \pi_1(X,x_0)\) where \(\gamma\) lifts to a path \(\tilde{\gamma}\) from \(\tilde{x}_0\) to \(\tilde{x}_1\). Thus \([\gamma]\) is in the normalizer \(N(H)\) iff \(p_*(\pi_1(\tilde{X},\tilde{x}_0))=p_*(\pi_1(\tilde{X},\tilde{x}_1))\), which by the lifting criterion is equivalent to the existence of a deck transformation taking \(\tilde{x}_0\) to \(\tilde{x}_1\). Hence the covering space is normal iff \(N(H) = \pi_1(X,x_0)\), that is, iff \(H\) is a normal subgroup of \(\pi_1(X,x_0)\).

Define \(\varphi : N(H) \rightarrow G(\tilde{X})\) sending \([\gamma]\) to the deck transformation \(\tau\) taking \(\tilde{x}_0\) to \(\tilde{x}_1\), in the notation above. Then \(\varphi\) is a homomorphism, for if \(\gamma'\) is another loop corresponding to the deck transformation \(\tau '\) taking \(\tilde{x}_0\) to \({x}_1'\) then \(\gamma \cdot \gamma'\) lifts to \(\tilde{\gamma} \cdot (\tau(\tilde{\gamma}'))\), a path from \(\tilde{x}_0\) to \(\tau(\tilde{x}_1')=\tau \tau'(\tilde{x}_0)\), so \(\tau \tau'\) is the deck transformation corresponding to \([\gamma][\gamma']\). By the preceding paragraph \(\varphi\) is surjective. Its kernel consists of classes \([\gamma]\) lifting to loops in \(\tilde{X}\). These are exactly the elements of \(p_*(\pi_1(\tilde{X},\tilde{x}_0))= H\). ◻

The group of deck transformations is a special case of the general notion of ‘groups acting on spaces’. Given a group \(G\) and a space \(Y\), then an action of \(G\) on \(Y\) is a homomorphism \(\rho\) from \(G\) to the group Homoe(\(Y\)) of all homeomorphisms from \(Y\) to itself. Thus to each \(g \in G\) is associated a homeomorphisms \(\rho(g): Y \rightarrow Y\), which for notational simplicity we write simply as \(g:Y \rightarrow Y\). For \(\rho\) to be a homomorphism amounts to requiring that \(g_1(g_2(y)) = (g_1g_2)(y)\) for all \(g_1,g_2 \in G\) and \(y \in Y\). If \(\rho\) is injective then it identifies \(G\) with a subgroup of \(\text{Homeo}(Y)\), and in practice not much is lost in assuming \(\rho\) is an inclusion \(G \hookrightarrow \text{Homeo}(Y)\) since in any case the subgroup \(\rho(G) \subset \text{Homeo}(Y)\) contains all the topological information about the action.

We shall be interested in actions satisfying the following condition:

(\({\Large *}\)) Each \(y \in Y\) has a neighborhood \(U\) such that all the images \(g(U)\) for varying

\(g \in G\) are disjoint. In other words, \(g_1(U) \cap g_2(U) \neq \emptyset\) implies \(g_1 = g_2\).

The action of the deck transformation group \(G(\tilde{x})\) on \(\tilde{X}\) satisfies (\({\Large *}\)). To see this. let \(\tilde{U} \subset \tilde{X}\) project homeomorphically to \(U \subset X\). If \(g_1(\tilde{U}) \cap g_2(\tilde{U}) \neq \emptyset\) for some \(g_1,g_2 \in G(\tilde{X})\), then \(g_1(\tilde{x}_1)=g_2(\tilde{x}_2)\) for some \(\tilde{x}_1,\tilde{x}_2\in \tilde{U}\). Since \(\tilde{x}_1\) and \(\tilde{x}_2\) must lie in the same set \(p^{-1}(x)\), which interesects \(\tilde{U}\) in only one point, we must have \(\tilde{x}_1=\tilde{x}_2\). THen \(g_1^{-1}g_2\) fixes this point, so \(g_1^{-1}g_2 = \mathbb{1}\) and \(g_1 = g_2\).

Note that in (\({\Large *}\)) it suffices to take \(g_1\) to be the identity since \(g_1(U) \cap g_2(U) \neq \emptyset\) is equivalent to \(U \cap g_1^{-1}g_2(U) \neq \emptyset\). Thus we have the equivalent condition that \(U \cap g(U) \neq \emptyset\) only when \(g\) is the identity.

Given an action of a group \(G\) on a space \(Y\), we can form a space \(Y/G\), the quotient space of \(Y\) in which each point \(y\) is identified with all its images \(g(y)\) as \(g\) ranges over \(G\). The points of \(Y/G\) are thus orbits \(Gy = \{g(y) \mid g \in G\}\) in \(Y\), and \(Y/G\) is called the orbit space of the action. For example, for a normal covering space \(\tilde{X} \rightarrow X\), the orbit space \(\tilde{X}/G(\tilde{X})\) is just \(X\).

Proposition 1.40. If an action of a group \(G\) on a space \(Y\) satisfies (\({\Large *}\)), then:

1. The quotient map \(p:Y \rightarrow Y/G,\, p(y)=Gy\), is a normal covering space.

2. \(G\) is the group of deck transformations of this covering space \(Y \rightarrow Y/G\) if \(Y\) is path-connected.

3. \(G\) is isomorphic to \(\pi_1(Y/G)/p_*(\pi_1(Y))\) if \(Y\) is path-connected and locally path- connected.

Proof: Given an open set \(U\subset Y\) as in condition (\({\Large *}\)), the quotient map \(p\) simply identifies all the disjoint homeomorphic sets \(\{g(U) \mid g \in G\}\) to a single open set \(p(U)\) in \(Y/G\). By the definition of the quotient topology on \(Y/G,\, p\) restricts to a homeomorphism from \(g(U)\) onto \(p(U)\) for each \(g \in G\) so we have a covering space. Each element of \(G\) acts as a deck transformation, and the covering space is normal since \(g_2g_1^{-1}\) takes \(g_1(U)\) to \(g_2(U)\). The deck transformation group contains \(G\) as a subgroup, and equals this subgroup if \(Y\) is path-connected, since if \(f\) is any deck transformation, then for an arbitrarily chosen point \(y \in Y\) and \(f(y)\) are in the same orbit and there is a \(g \in G\) with \(g(y) = f(y)\), hence \(f=g\) since deck transformations of a path-connected covering space are uniquely determined by where they send a point. The final statement of the proposition is immediate from part (b) of Proposition 1.39. ◻

In view of the preceding proposition, we shall call an action satisfying (\({\Large *}\)) a covering space action. This is not standard terminology, but there does not seem to be a universally accepted name for actions satisfying (\({\Large *}\)). Sometimes these are called ‘properly discontinuous’ actions, but more often this rather unattractive term means something weaker: Every point \(x \in X\) has a neighborhood \(U\) such that \(U \cap g(U)\) is nonempty for only finitely many \(g \in G\). Many symmetry groups have this proper discontinuity property without satisfying (\({\Large *}\)), for example the group of symmetries of the familiar tiling of \(\mathbb{R}^2\) by regular hexagons. The reason why the action of this group on \(\mathbb{R}^2\) fails to satisfy (\({\Large *}\)) is that there are fixed points: points \(y\) for which there is a nontrivial element \(g \in G\) with \(g(y)=g\). For example, the vertices of the ehxagons are fixed by the \(120\) degree rotations abotu these points, and the midpoints of edges are fixed by \(180\) degree rotations. An action without fixed points is called a free action. Thus for a free action of \(G\) on \(Y\), only the identity element of \(G\) fixes any point of \(Y\). This is equivalent to requiring that all the images \(g(y)\) of each \(y \in Y\) are distinct, or in other owrds \(g_1(y)=g_2(y)\) only when \(g_1 = g_2\), since \(g_1(y)=g_2(y)\) is equivalent to \(g^{-1}_1g_2(y)=y\). Though condition (\({\Large *}\)) implies freeness, the converse is not always true. An example is the action of \(\mathbb{Z}\) on \(S^1\) in which a generator of \(\mathbb{Z}\) acts by rotation through an angle \(\alpha\) that is an irrtional multiple of \(2\pi\). In this case each orbit \(\mathbb{Z}y\) is dense in \(S^1\), so condition (\({\Large *}\)) cannot hold since it implies that orbits are discrete subspaces. An exercise at the end of the section is to show that for actions on Hausdorff spaces, freeness plus proper discontinuity implies condition (\({\Large *}\)). Note that proper discontinuity is automatic for actions by a finite group.

Example 1.41. Let \(Y\) be the closed orientable surface of genus \(11\), an ‘\(11\)-hole torus’ as shown in the figure. This has a \(5\)-fold rotational symmetry, generated by a rotation of angle \(2\pi / 5\). Thus we have the cyclic group \(\mathbb{Z}_5\) acting on \(Y\), and the condition (\({\Large *}\)) is obviously satisfied. The quotient space \(Y/\mathbb{Z}_5\) is a surface of genus \(3\), obtained from one of the five subsrufaces of \(Y\) cut off by the circles \(C_1, \cdots, C_5\) by identifying its two boundary circles \(C_i\) and \(C_{i+1}\) to form the circle \(C\) as shown. Thus we have a covering space \(M_{11} \rightarrow M_3\) where \(M_g\) denotes the closed orientable surface of genus \(g\). In particular, we see that \(\pi_1(M_3)\) contains the ‘larger’ group \(\pi_1(M_{11})\) as a normal subgroup of index \(5\), with quotient \(\mathbb{Z}_5\). This example obviously generalizes by replacing the two holes in each ‘arm’ of \(M_{11}\) by \(m\) holes and the \(5\)-fold symmetry by \(n\)-fold symmetry. This gives a covering space \(M_{mn+1} \rightarrow M_{m+1}\). An exercise in §2.2 is to show by an Euler characteristic argument that if there is a covering space \(M_g \rightarrow M_g\) then \(g=mn+1\) and \(h=m+1\) for some \(m\) and \(n\).

As a special case of the final statement of the preceding proposition we see that for a covering space action of a group \(G\) on a simply-connected locally path-connected space \(Y\), the orbit spacer \(Y/G\) has fundamental group isomorphic to \(G\). Under this isomorphism an element \(g \in G\) corresponds to a loop in \(Y/G\) that is the projection of a path in \(Y\) from a chosen basepoint \(y_0\) to \(g(y_0)\). Any two such paths are homotopic since \(Y\) is simply-connected, so we get a well0defined element of \(\pi_1(y/G)\) associated to \(g\).

This method for computing fundamental groups vai group actions on simply-connected spaces is essentially how we computed \(\pi_1(S^1)\) in §1.1, via the covering space \(\mathbb{R} \rightarrow S^1\) arising from the action of \(\mathbb{Z}\) on \(\mathbb{R}\) by translations. This is a useful general technique for computing fundamental groups, in fact. Here are some examples illustrating this idea.

Example 1.42. Consider the grid in \(\mathbb{R}^2\) formed by the horizontal and vertical lines through points in \(\mathbb{Z}^2\). Let us decorate this grid with arrows in either of the two ways shown in the figure, the difference between the two cases being that in the second case the horizontal arrows in adjacent lines point in opposite directions. The group \(G\) consisting of all symmetries of the first decorated grid is isomorphic to \(\mathbb{Z} \times \mathbb{Z}\) since it consists of all translations \((x,y) \mapsto (x+m,y+n)\) for \(m,n \in \mathbb{Z}\). For the second grid the symmetry group \(G\) contains a subgroup of translations of the form \((x,y \mapsto (x+m, y+2n))\) for \(m,n \in \mathbb{Z}\), but there are also glide-reflection symmetries consisting of vertical translation by an odd integer distance followed by reflection across a vertical line, either a vertical line of the grid or vertical line halfway between two adjacent grid lines. For both decorated grids there are elements of \(G\) taking any square to nay other, butr only the identity element of \(G\) takes a square to itself. The minimum distance any point is moved by a nontrivial element of \(G\) is \(1\), which easily implies the covering space condition (\({\Large *}\)). The orbit space \(\mathbb{R}^2/G\) is the quotient space of a square in the grid with opposite edges identified according to the arrows. Thus we see that the fundamental groups of the torus and the Klein bottle are the symmetry groups \(G\) in the two cases. In the second case the subgroup of \(G\) formed by the translations ahs index two, and the orbit space for this subgroup is a torus forming a two-sheeted covering spaces of the Klein bottle.

Example 1.43: \(\mathbb{R}P^n\). The antipodal map of \(S^n,\, x\mapsto -x\), generates an action of \(\mathbb{Z}_2\) on \(S^n\) with orbit space \(\mathbb{R}P^n\), real projective \(n\)-space, as defined in Example 0.4. The action is a covering space action since each open hemisphere in \(S^n\) is disjoint from its antipodal image. As we saw in Proposition 1.14, \(S^n\) is simply-connected if \(n \geq 2\), so from the covering space \(S^n \rightarrow \mathbb{R}P^n\) we deduce that \(\pi_1(\mathbb{R}P^n) \approx \mathbb{Z}_2\) for \(n \geq 2\). A generator for \(\pi_1(\mathbb{R}P^n)\) is any loop obtained by projecting a path in \(S^n\) connecting two antipodal points. One can see explicitly that such a loop \(\gamma\) has order two in \(\pi_1(\mathbb{R}P^n)\) if \(n \geq 2\) since the composition \(\gamma \cdot \gamma\) lifts to a loop in \(S^n\), and this can be homotoped to the trivial loop since \(\pi_1(S^n) = 0\), so the projection of this homotopy into \(\mathbb{R}P^n\) gives a nullhomotopy of \(\gamma \cdot \gamma\).

One may ask whether there are other finite groups that act freely on \(S^n\), defining covering spaces \(S^n \rightarrow S^n/G\). We will show in Proposition 2.29 that \(\mathbb{Z}_2\) is the only possibility when \(n\) is even, but for odd \(n\) the question is much more difficult. It is easy to construct a free action of any cyclic group \(\mathbb{Z}_m\) on \(S^{2k-1}\), the action generated by the rotation \(v \mapsto e^{2\pi i/m}v\) of the unit sphere \(S^{2k-1}\) in \(\mathbb{C}^k = \mathbb{R}^{2k}\). This action is free since an equation \(v=e^{2\pi i/m}v\) with \(0 < l < m\) implies \(v=0\), but \(0\) is not a point of \(S^{2k-1}\). The orbit space \(S^{2k-1}/\mathbb{Z}_m\) is one of a family of spaces called lens spaces defined in Example 2.43.

THere are also noncylic finite groups that act freely as rotations of \(S^n\) for odd \(n>1\). These actions are classified quite explicitly in [Wolf 1984]. Examples in the simplest case \(n=3\) can be produced as follows. View \(\mathbb{R}^4\) as the quaternion algebra \(\mathbb{H}\). Multiplication of quaternions satisfies \(|ab| = |a||b|\) where \(|a|\) denotes the usual Euclidean length of a vector \(a \in \mathbb{R}^4\). Thus if \(a\) and \(b\) are unit vectors, so is \(ab\), and hence quaternion multiplication defines a map \(S^3 \times S^3 \rightarrow S^3\). This in fact makes \(S^3\) into a group, though associativity is all we need now since associativity implies that any subgroup \(G\) of \(S^3\) acts on \(S^3\) by left-multiplication, \(g(x)=gx\). This action is free since an equation \(x=gx\) in the division algebra \(\mathbb{H}\) implies \(g=1\) or \(x=0\). As a concrete example, \(G\) could be the familiar quaternion group \(Q_8 = \{\pm1, \pm i, \pm h, \pm k\}\) from group theory. More generally, for a positive integer \(m\), let \(Q_{4m}\) be the subgroup of \(S^3\) generated by the two quaternions \(a=e^{\pi i/m}\) and \(b=j\). Thus \(a\) has order \(2m\) and \(b\) has order \(4\). The easily verified relations \(a^m=b^2=-1\) and \(bab^{-1}=a^{-1}\) imply that the subgroup \(\mathbb{Z}_{2m}\) generated by \(a\) is normal and of index \(2\) in \(Q_{4m}\). Hence \(Q_{4m}\) is a group of order \(4m\), called generalized quaternion group. Another common name for this group is the binary dihedral group \(D^*_{4m}\) since its quotient by the subgroup \(\{\pm 1\}\) is the ordinary dihedral group \(D_{2m}\) of order \(2m\).

Beside the groups \(Q_{4m}=D^*_{4m}\) there are just three other noncyclic finite subgroups of \(S^3\): the binary tetrahedral, octahedral, and icosahedral groups \(T^*_{24},\, O^*_{48}\), and \(I^*_{120}\), of orders indicated by the subscripts. These project two-to-one onto the groups of rotational symmetries of a regular tetrahedron, octahedron (or cube), and icosahedron (or dodecahedron). In fact, it is not hard to see that the homomorphism \(S^3 \rightarrow SO(3)\) sending \(u \in S^3 \subset \mathbb{H}\) to the isometry \(v \rightarrow u^{-1}vu\) of \(\mathbb{R}^3\), viewing \(\mathbb{R}^3\) as the ‘pure imaginary’ quaternions \(v=ai+bj+ck\), is surjective with kernel \(\{\pm 1\}\). Then the groups \(D^*_{4m},\,T^*_{24},\, O^*_{48},\,I^*_{120}\) are the preimages in \(S^3\) of the groups of rotational symmetries of a regular polygon or polyhedron in \(\mathbb{R}^3\).

There are two conditions that a finite group \(G\) acting freely on \(S^n\) must satisfy:

1. Every abelian subgroup of \(G\) is cyclic. This is equivalent to saying that \(G\) contains no subgroup \(\mathbb{Z}_p \times \mathbb{Z}_p\) with \(p\) prime.

2. \(G\) contains at most one element of order \(2\).

A proof of (a) is sketched in an exercise for §4.2. For a proof of (b) the original sourcce [Milnor 1957] is recommended reading. The groups satisfying (a) have been completely classified; see [Brown 1982], section VI.9, for details. An example of a group satisfying (a) but not (b) is the dihedral group \(D_{2m}\) for odd \(m > 1\).

There is also a much more difficult converse: A finite group satisfying (a) and (b) acts freely on \(S^n\) for some \(n\). References for this are [Madse, Thomas, & Wall 1976] and [Davis & Milgram 1985]. There is also almost complete information about which \(n\)’s are possible for a given group.

Example 1.44. In Example 1.35 we constructed a contractible \(2\)-complex \(\tilde{X}_{m,n}=T_{m,n} \times \mathbb{R}\) as the universal cover of a finnite \(2\)-complex \(X_{m,n}\) that was the union of themapping cylinders of the two maps \(S^1 \rightarrow S^1,\,z\mapsto Z^m\) and \(z\mapsto z^n\). The group of deck transformations of this covering space is therefore the fundamental group \(\pi_1(X_{m,n})\). From van Kampen’s theorem applied to the decomposition of \(X_{m,n}\) into the two mapping cylinders we have the presentation \(\langle a,b \mid a^m b^{-n} \rangle\) for this group \(G_{m,n}=\pi_1(X_{m,n})\). It is interesting to look at the action of \(G_{m,n}\) on \(\tilde{X}_{m,n}\) more closely. We described a decomposition of \(\tilde{X}_{m,n}\) into rectangles, with \(X_{m,n}\) the quotient of one rectangle. These rectangles in fact define a cell structure on \(\tilde{X}_{m,n}\) lifting a cell structure on \(X_{m,n}\) with two vertices, three edges, and one \(2\)-cell. The group \(G_{m,n}\) is thus a group of symmetries of this cell structure on \(\tilde{X}_{m,n}\), then \(G_{m,n}\) is the group of all symmetries of \(\tilde{X}_{m,n}\) preserving the orientations of edges. For example, the element \(a\) acts as a ‘screw motion` about an axis that is a vertical line \(\{v_a\} \times \mathbb{R}\) with \(v_a\) a vertex of \(T_{m,n}\), and \(b\) acts similarly for a vertex \(v_b\).

Since the action of \(G_{m,n}\) on \(\tilde{X}_{m,n}\) preserves the cell structure, it also preserves the product structure \(T_{m,n} \times \mathbb{R}\). This means that there are actions of \(G_{m,n}\) on \(T_{m,n}\) and \(\mathbb{R}\) such that the action on the product \(X_{m,n}=T_{m,n} \times \mathbb{R}\) is the diagonal action \(g(x,y)=(g(x),g(y))\) for \(g \in G_{m,n}\). If we make the rectangles of unit height in the \(\mathbb{R}\) coordinate, then the element \(a^m=b^n\) acts on \(\mathbb{R}\) as unit translation, while \(a\) acts by \(\frac{1}{m}\) translation and \(b\) by \(\frac{1}{n}\) translation. The translation actions of \(a\) and \(b\) on \(\mathbb{R}\) generate a group of translations of \(\mathbb{R}\) that is infinite cyclic, generated by translation by the reciprocal of the least common multiple of \(m\) and \(n\).

The action of \(G_{m,n}\) on \(T_{m,n}\) has kernel consisting of the powers of the element \(a^m=b^n\). This infinite cyclic subgroup is precisely the center of \(G_{m,n{\), as we saw in Example 1.24. There is an induced action of the quotient group \(\mathbb{Z}_m * \mathbb{Z}_n\) on \(T_{m,n}\), but this is not a free action since the elements \(a\) and \(b\) and all their conjugates fix vertices of \(T_{m,n}\). On the other hand, if we restrict the action of \(G_{m,n}\) on \(T_{m,n}\) to the kernel \(K\) of the map \(G_{m,n} \rightarrow \mathbb{Z}\) given by the action of \(G_{m,n}\) on the \(\mathbb{R}\) factor of \(X_{m,n}\), then we do obtain a free action of \(K\) on \(T_{m,n}\). Since this action takes vertices to vertices and edges to edges, it is a covering space action, so \(K\) is a free group, the fundamental group of the graph \(T_{m,n}/K\). An exercise at the end of the section is to deterimne \(T_{m,n}/K\) explicitly and compute the number of generators of \(K\).

Cayley Complexes

Covering spaces can be used to describe a very classical method for viewing groups geometrically as graphs. Recall from Corollary 1.28 how we associated to each group presentation \(G=\langle g_\alpha \mid r_\beta \rangle\) a \(2\)-dimensional cell complex \(X_G\) with \(\pi_1(X_G) \approx G\) by taking a wedge-sum of circles, one for each generator \(g_\alpha\), and then attaching a \(2\)-cell for each relator \(r_\beta\). We can construct a cell complex \(\tilde{X}_G\) with a covering space action of \(G\) such that \(\tilde{X}_G/G=X_G\) in the following way. Let the vertices of \(\tilde{X}_G\) be the elements of \(G\) themselves. Then, at each vertex \(g \in G\), insert an edge joining \(g\) to \(gg_\alpha\) for each of the chosen generators \(g_\alpha\). The resulting graph is known as the Cayley graph of \(G\) with respect to the generators \(g_\alpha\). This graph is connected since every element of \(G\) is a product of \(g_\alpha\)’s, so there is a path in the graph joining each vertex to the identity vertex \(e\), Each relation \(r_\beta\) determines a loop in the graph, starting at any vertex \(g\), and we attach a \(2\)-cell for each such loop. The resulting cell complex \(\tilde{X}_G\) is the Cayley complex of \(G\). The group \(G\) acts on \(\tilde{X}_g\) by multiplication on the left. Thus, an element \(g \in G\) sends a vertex \(g' \in G\) to the vertex \(gg'\), and the edge from \(g'\) to \(g'g_\alpha\) is sent to the edge from \(gg'\) to \(gg'g_\alpha\) The action extends to \(2\)-cells in the obvious way. This is clearly a covering space action, and the orbit space is just \(X_G\).

In fact \(\tilde{X}_G\) is the universal cover of \(X_G\) since it is simply-connected. This can be seen by considering the homomorphims \(\varphi : \pi_1(X_G) \rightarrow G\) defined in the proof of Proposition 1.39. For an edge \(e_\alpha\) in \(X_G\) corresponding to a generator \(g_\alpha\) of \(G\), it is clear from the definition of \(\varphi\) that \(\varphi ([e_\alpha])=g_\alpha\), so \(\varphi\) is an isomorphism. In particular the kernel of \(\varphi,\,p_*(\pi_1(\tilde{X}_G))\), is zero, hence also \(\pi_1(\tilde{X}_G)\) since \(p_*\) is injective.

Let us look at some examples of Cayley complexes.

Example 1.45. When \(G\) is the free group on two generators \(a\) and \(b\), \(X_G\) is \(S^1 \vee S^1\) and \(\tilde{X}_G\) is the Cayley graph of \(\mathbb{Z} * \mathbb{Z}\) pictured at the right. The action of \(a\) on this graph is a rightward shift along the central horizontal axis, while \(b\) acts by an upward shift along the central vertical axis. The composition \(ab\) of these two shifts then takes the vertex \(e\) to the vertex \(ab\). Similarly, the action of any \(w \in \mathbb{Z} * \mathbb{Z}\) takes \(e\) to the vertex \(w\).

Example 1.46. The group \(G=\mathbb{Z} \times \mathbb{Z}\) with presentation \(\langle x,y \mid xyx^{-1}y^{-1} \rangle\) has \(X_G\) the torus \(S^1 \times S^1\), and \(\tilde{X}_G\) is \(\mathbb{R}^2\) with vertices the integer lattice \(\mathbb{Z}^2 \subset \mathbb{R}^2\) and edges the horizontal and vertical segments between these lattice points. The action of \(G\) is by translations \((x,y) \mapsto (x+m,y+n)\).

Example 1.47. For \(G=\mathbb{Z}_2= \langle x \mid x^2 \rangle ,\, X_G\) is \(\mathbb{R}P^2\) and \(\tilde{X}_G = S^2\). More generally, for \(\mathbb{Z}_n = \langle x \mid x^n \rangle ,\, X_G\) is \(S^1\) with a disk attached by the map \(z \mapsto z^n\) and \(\tilde{X}_G\) consists of \(n\) disks \(D_1 , \cdots , D_n\) with their boundary circles identified. A generator of \(\mathbb{Z}_n\) acts on this union of disks by sending \(D_i\) to \(D_{i+1}\) via a \(2\pi / n\) rotation, the subscript \(i\) being taken mod \(n\). The common boundary circle of the disks is rotated by \(2\pi / n\).

Example 1.48. If \(G=\mathbb{Z}_2 * \mathbb{Z}_2 = \langle a,b \mid a^2, b^2 \rangle\) then the Cayley graph is a union of an infinite sequence of circles each tangent to its two neighbors.

We obtain \(\tilde{X}_G\) from this graph by making each circle the equator of a \(2\)-sphere, yielding an infinite sequence of tangent \(2\)-spheres. Elements of the index-two normal subgroup \(\mathbb{Z} \subset \mathbb{Z}_2 * \mathbb{Z}_2\) generated by \(ab\) act on \(\tilde{X}_G\) as translations by an even number of units, while each of the remaining elements of \(\mathbb{Z}_2 * \mathbb{Z}_2\) acts as the antipodal map on one of the spheres and flips the whole chain of spheres end-for-end about this sphere. The orbit space \(X_G\) is \(\mathbb{R}P^2 \vee \mathbb{R}P^2\)

It is not hard to see the generalization of this example to \(\mathbb{Z}_m * \mathbb{Z}_n\) with the presentation \(\langle a,b \mid a^m, b^n \rangle\), so that \(\tilde{X}_G\) consists of an infinite union of copies of the Cayley complexes for \(\mathbb{Z}_m\) and \(\mathbb{Z}_n\) constructed in Example 1.47, arranged in a tree-like pattern. The case of \(\mathbb{Z}_2 * \mathbb{Z}_3\) is pictured below.