DEGREE-REGULAR TRIANGULATIONS OF SURFACES BASUDEB DATTA AND SUBHOJOY GUPTA Abstract. A degree-regular triangulation is one in which each vertex has iden- tical degree. Our main result is that any such triangulation of a (possibly non- compact) surface S is geometric, that is, it is combinatorially equivalent to a geodesic triangulation with respect to a constant curvature metric on S, and we list the possibilities. A key ingredient of the proof is to show that any two d-regular triangulations of the plane for d > 6 are combinatorially equivalent. The proof of this uniqueness result, which is of independent interest, is based on an inductive argument involving some combinatorial topology. 1. Introduction A triangulation of a surface S is an embedded graph whose complementary regions are faces bordered by exactly three edges (see x2 for a more precise definition). Such a triangulation is d-regular if the valency of each vertex is exactly d, and geometric if the triangulation are geodesic segments with respect to a Riemannian metric of constant curvature on S, such that each face is an equilateral triangle. Moreover, two triangulations G1 and G2 of X are combinatorially equivalent if there is a homeomorphism h : X ! X that induces a graph isomorphism from G1 to G2. The existence of such degree-regular geometric triangulations on the simply- connected model spaces (S2; R2 and H2) is well-known. Indeed, a d-regular geomet- ric triangulation of the hyperbolic plane H2 is generated by the Schwarz triangle group ∆(3; d) of reflections across the three sides of a hyperbolic triangle (see x3.2 for a discussion). In this note we prove: Theorem 1.1. Any d-regular triangulation of a surfaceS is combinatorially equiv- alent to a geometric triangulation, and one of the following three possibilities hold: • d < 6, S ∼= S2 and the triangulation corresponds to the boundary of the ∼ 2 tetrahedron, octahedron or icosahedron, or S = RP and the triangulation is a two-fold quotient of the boundary of the icosahedron, • d = 6, the universal cover S~ = E2 and the lift of the triangulation is the standard hexagonal triangulation, • d > 6, S~ = E2 and the lift of the triangulation is the geodesic triangulation generated by the Schwarzian triangle group ∆(3; d). 1 2 BASUDEB DATTA AND SUBHOJOY GUPTA In the case when S is compact, this was proved in [EEK82]; indeed, our work also gives a new proof of their result. A key ingredient in our proof of Theorem 1.1 is the following uniqueness result for triangulations of the plane: Theorem 1.2. For each integer d > 6, any two d-regular triangulations of R2 are combinatorially equivalent. In previous work [DU05] (see also [DM17]), it was shown that the above result is true for d = 6 as well; in this case the triangulation is combinatorially equivalent to the familiar hexagonal triangulation of the Euclidean plane E2. This paper develops a different technique that works for the case of a degree- regular triangulation of the hyperbolic plane H2, where the number of vertices in a compact subcomplex grows exponentially with the radius (see Lemma 4.5). Our proof involves an inductive procedure to construct an exhaustion of the plane by compact triangulated regions; we start with the link of a single vertex, and at each step, add all the links of the boundary vertices. A crucial part of the argument is to ensure that at each step, the region we obtain is a triangulated disk that is determined uniquely by the preceding step; this needs a topological argument involving the Euler characteristic. A uniqueness result akin to Theorem 1.2 is the uniqueness of Archimedean tilings, which was recently proved in [DM] and forms the inspiration for the present work. Note that a tiling (or map) of a surface is a more general notion when the embed- ded graph has complementary faces that are (topological) polygons. The type of a vertex in a tiling is the cyclic collection of integers that determines the number of sides of each polygon surrounding it. If all the vertices in a tiling are of same type then the tiling is called a semi-equivelar map or a uniform tiling. An Archimedean tiling of the plane R2 is a semi-equivelar map of R2 by regular Euclidean polygons; it is a classical fact that there are exactly 11 such tilings of R2 (see also [GS77]). The techniques of this paper can be extended to show the uniqueness of equivelar maps, which are uniform tilings where each vertex is of type [pq], that is, each tile is a p-gon and there are exactly q such polygons around each vertex. This would imply that any equivelar map of a surface is geometric. Once again, this was shown by [EEK82], but only in the case when S is compact. However the following question is, to the best of our knowledge, still open: Question 1.3. Are any two uniform tilings of R2 with the same vertex type com- binatorially equivalent? We hope to address this in a subsequent article. DEGREE-REGULAR TRIANGULATIONS OF SURFACES 3 2. Preliminaries We begin by setting up some definitions and notation that we shall use throughout the paper. A standard reference for basic terminology on graphs is [BM08]. For a graph G, V (G) and E(G) will denote its vertex-set and edge-set respectively, Definition 2.1 (Triangulation). A triangulation of a surface S is an embedded graph G such that • all complementary regions (faces) are (topological) triangles, • each edge in E(G) belongs to (the closure of) exactly two triangles, • each vertex in V (G) belongs to at least 3 triangles, arranged in a cyclic order around it. Further, for v 2 V (G), dG(v) denotes the degree of v in G. A triangulation is d-regular for d ≥ 3 if moreover, each vertex has valency (degree) d. A path with edges u1u2; : : : ; un−1un is denoted by u1-u2- ··· -un. For n ≥ 3, an n-cycle with edges u1u2; : : : ; un−1un; unu1 is denoted by u1-u2- ··· -un-u1 and also by Cn(u1; u2; : : : ; un). If G is a graph and u is a vertex not in G then the join u ∗ G is the simplicial complex whose vertex set is V (G) t fug, edge-set is E(G) t fuv : v 2 V (G)g and the set of 2-simplices is fuvw : vw 2 E(G)g. By a triangulated surface, we mean a simplicial complex whose underlying topo- logical space is a surface (possibly with boundary), such that its 1-skeleton is a triangulation as defined above. Since a closed surface can not be a proper subset of a surface, it follows that a closed triangulated surface can't be a proper sub- complex of a triangulated surface. We identify a triangulated surface by the set of 2-simplices in the complex. If u is a vertex in a 2-dimensional simplicial complex X then the link of u in X, denoted by lkX (u) (or simply lk(u)), is the largest subgraph G of X such that u 62 G and u ∗ G ⊆ X. The subcomplex u ∗ lkX (u) is called the star of u in X and is denoted by stX (u) or simply by st(u). Clearly, the star of a vertex in a triangulated surface is a triangulated 2-disc. A 2-dimensional simplicial complex is a triangulated surface if and only if the links of vertices in X are paths and cycles. Moreover, for a triangulated surface X, lkX (u) is a path if and only if u is a boundary vertex of X. If not mentioned otherwise, by a triangulated surface we mean a triangulated surface without boundary. 4 BASUDEB DATTA AND SUBHOJOY GUPTA 3. Geometric triangulations of surfaces In this section we discuss constructions of d-regular triangulated surfaces where the triangulations are geometric. 3.1. Case of degree d < 6. The following result lists all possibilities of a d- regular triangulated surface when the degree d < 6. Lemma 3.1. Let X be a triangulated surface without boundary. (i) If X is 3-regular, then X is the boundary of a tetrahedron. (ii) If X is 4-regular, then X is the boundary of an octahedron. 2 (iii) If X is 5-regular, then X is either the boundary of an icosahedron, or RP6 (the minimal 6-vertex triangulation of the real projective plane). Remark 3.2. Note that in the following proof of Lemma 3.1 we do not assume that X is finite. 1 0 !a 0 ! %e a ¡SA !! aa A 5 a! % e 4 !a 2 ¡ S aa% 3 e!! ¡ AAS %e %e ¡ AS % e % e ¡ A S 2 % e% e 5 ¡ A S b " A b 0 " ¡ S b " 10 A b " ¡ !a A S b" ¡ ! %e a S 1 0 !! aa A 0 2 5 a!¡ % e 4 !aA 2 S RP6 aa 3 e!! @ S % e %e % @ S % e % e @ S 2 % e% e 5 @ S "b "b " b 0 " b @ S " b " b " b " b @ S " b" b @S " `` `` b " 1 `` `` b@S " `` ` b 0 " ` `` b@S 0 4 " `` `b@S 3 @ I Figure 1 Proof. Part (i). Let a be a vertex whose link is C3(b; c; d). Then c-a-d ⊆ lkX (b). Since, deg(b) = 3, it follows that lkX (b) = C3(c; a; d). Thus, bcd is a simplex and hence the boundary @∆ of the tetrahedron ∆ = abcd is a subcomplex of X. Since, @∆ is a closed triangulated surface, it follows that X = @∆.