Locally symmetric spaces play an important part in differential geometry and arise from many different areas such as topology, number theory, representation theory, algebraic geometry,.The typical important class consists of quotients of symmetric spaces by arithmetic groups, for example, the moduli space of elliptic curves is the quotient of the upper half plane H2 by SL(2, Z). | JOURNAL OF SCIENCE, Hue University, , , 2011 SOME RESULTS ON THE LOCALLY SYMMETRIC STRUCTURE OF HALF UPPER SPACES 2 Tran Dao Dong1 and Hoang Thai Vu2 1 Hue University Department of Education and Training, Thua Thien Hue Province Abstract. Locally symmetric spaces play an important part in differential geometry and arise from many different areas such as topology, number theory, representation theory, algebraic geometry,.The typical important class consists of quotients of symmetric spaces by arithmetic groups, for example, the moduli space of elliptic curves is the quotient of the upper half plane H2 by SL(2, Z). In this paper, firstly, we study the symmetric structure of the upper half space H3 and the relation with the symmetric space SL(2, C)/SU (2). Then we study the locally symmetric space SL(2, Z + iZ)\H3 ∼ = SL(2, Z + iZ)\SL(2, C)/SU (2) based on the action of the discrete group SL(2, Z + iZ) on H3 . 1 The structure of the quotient SL(2, Z)\H2 Let H2 = {x + yi|x, y ∈ R, y > 0} be the Poincaré half plane. Then, as in [5], based on the transitive isometric action of the Lie group SL(2, R) on H2 , we can show easily that H2 is isomorphic with the symmetric space G/K = SL(2, R)/SO(2, R). Moreover, consider the modular group Γ = SL(2, Z), we see that the locally symmetric space associated with the triple (G, K, Γ) is the quotient SL(2, Z)\H2 . Now we study the structure of the quotient SL(2, Z)\H2 . For this purpose, we need to introduce the notion of fundamental domains. Definition ([5, Definition ]). A fundamental domain for a discrete group Γ acting on H2 is an open subset Ω ⊂ H2 such that i) Each coset contains at least one point in the closure Ω; that is, H2 = ; ii) No two interior points of Ω lie in one Γ−orbit; that is, γΩ, γ ∈ Γ, are disjoint open subsets. Given a fundamental domain Ω, we can find a subset F, Ω ⊂ F ⊂ Ω, such that each Γ−orbit contains exactly one point in F, and there is a bijective map from Γ\H2 to .
