It is known that a closed (ie. connected, compact without boundary) surface is homeomorphic to a sphere or to a connected sum of finitely-many tori or projective spaces. Let (resp. ) denote the connected sum of tori (resp. projective spaces). Because , we easily deduce the Euler characteristics and ; the integer is called the genus.
A surface group is a group isomorphic to the fundamental group of one of these surfaces. If is a surface group, two cases happens: either is the fundamental group of an orientable surface of genus and
or is the fundamental group of a non-orientable surface of genus and
(Just write or as a polygon with pairwise identified edges, and apply van Kampen’s theorem; for more information, see Massey’s book Algebraic Topology.)
Our main result is the following characterization of subgroups of a surface group:
Theorem 1: Let be the fundamental group of a closed surface and be a subgroup. Either has finite index in and is isomorphic to the fundamental group of a closed surface whose Euler characteristic is , or is an infinite-index subgroup of and is free.
For this note, I was inspired by Jaco’s article, On certain subgroups of the fundamental group of a closed surface. The proof of property 2 below comes from Stillwell’s book, Classical topology and combinatorial group theory.
Essentially, the theorem above follows from Property 2 below:
Property 2: The fundamental group of a non-compact surface is free.
Combined with Property 3, let us first notice some corollaries of Theorem 1 on the structure of surface groups.
Property 3: The abelianization of (resp. ) is isomorphic to (resp. ). Therefore, two closed surfaces and are homeomorphic if and only if their fundamental groups are isomorphic.
Proof. In order to compute the abelianizations, it is sufficient to add all the possible commutators as relations in the presentations given above. Therefore,
and, with ,
Consequently, the and define a family of pairwise non-isomorphic groups, and the conclusion follows from the classication of closed surfaces.
Corollary 1: Let be the fundamental group of a closed surface different from the projective space. Then is torsion-free.
Proof. The abelianization of a surface group is cyclic if and only if it is the fundamental group of the projective plane; therefore, the same conclusion holds for the surface groups themself, and we deduce that has no cyclic subgroup of finite index. Consequently, the cyclic subgroup generated by a non trivial element of is free, that its order is infinite.
Corollary 2: Let be the fundamental group of a closed surface satisfying and be a subgroup generated by elements. If then is free.
Proof. If , is a torus or a Klein bottle, and the statement just says that its fundamental group is torsion-free, that follows from Corollary 1.
From now on, we suppose that . According to Property 3, the abelianization of has rank . We deduce that the smallest cardinality of a generating set of has cardinality ; in particular, because . Suppose that has finite index . Then is the fundamental group of a closed surface and . In the same way, necessarily . Therefore,
Corollary 3: Let be the fundamental group of a closed surface satisfying . If commute, then there exist and such that and .
Proof. Let . Then it is easy to find two epimorphisms
Therefore, the groups and are not abelian for , and we deduce that the sphere, the projective plane and the torus are the only closed surfaces whose fundamental group is abelian.
Because satisfies , we conclude that has no finite-index abelian subgroup, and that the subgroup genereted by is necessarily free; since and commute, the subgroup turns out to be cyclic and the conclusion follows.
Corollary 4: The commutator subgroup of a surface group is free.
Proof. Let be the fundamental group of a closed surface . If is a projective space, then and its commutator subgroup is trivial (in particular free). Otherwise, thanks to Property 3 we know that the abelianization of is infinite; therefore, the commutator subgroup is an infinite-index subgroup and so is free.
Proof of property 2. First, we notice that the fundamental group of a connected compact surface with boundary is free. If is such a surface, by gluing a disk along each boundary component, we get a closed surface ; therefore, is homotopic to a punctured closed surface. Because a closed surface may be identified with a polygon whose edges are pairwise identified, it is not difficult to prove that a punctured closed surface is homotopic to a graph (by “enlarging the holes”); in particular, we deduce that is free.
The figure below shows that the fundamental group a 2-punctured torus is a free group of rank three.
From now on, let be a non-compact surface. In order to prove that is free, we want to find a sequence of compact surfaces with boundary
such that and that the inclusions are -injective. It is sufficient to conclude since we proved above that the fundamental group of a compact surface with boundary is free.
Take a triangulation of so that may be identified with a simplicial complex; let denote the 2-simplexes. Without loss of generality, we may suppose that is adjacent to one of the simplexes ; otherwise, number the simplexes by taking first the simplexes adjacent to a vertex cyclically, then the simplexes within from , then the simplexes within from , etc.
Notice that a connected union of simplexes may not be a surface; however, if denotes the union of with a small closed ball around each of its vertices, a connected union of is automatically a compact surface with boundary. Now, we construct the surfaces by induction:
Let . Now suppose that is given. If , then set ; otherwise, define as the union of with and with the disks bounding a boundary component of .
Because is a union of , we know that it is a compact surface with boundary. To conclude, it is sufficent to prove that the inclusion is -injective. To do that, just notice that is attatched on the boundary of in six possible ways:
In the first three cases, because the boundary component bounds a disk in if and only if it bounds a disk in . So clearly retracts on so that the inclusion induces an isomorphism .
In the three last cases, up to homotopy, we just glue one or two edges on a boundary component, and using van Kampen’s theorem, we notice that a free basis of is obtained by adding one or two elements to a free basis of .
Proof of theorem 1. Let be the covering associated to the subgroup ; in particular, is a surface of fundamental group . If is a finite-index subgroup, is compact and the conclusion follows. Otherwise, is not compact, and the conclusion follows from Property 2.