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ERRATUM TO THE PAPER ‘INTEGRALITY AND ARITHMETICITY OF CO-COMPACT LATTICE CORRESPONDING TO CERTAIN COMPLEX TWO-BALL QUOTIENTS OF PICARD NUMBER ONE’, ASIAN J. MATH., 8(2004) 107-130

Lemma 1. The set f ◦ p̃(Ha) is convex. Proof Suppose Σ is an apartment of the building such that f ◦ p̃(Ha) ∩ Σ is unbounded. We need to show that the image f ◦ p̃(Ha) ∩ Σ is isometric to R ∩ Σ as a set with the Euclidean metric, if R ∩ Σ #= ∅, where Ha = h̃−1(a). The earlier discussions in the article shows that it is true for each chamber in Σ. Without loss of generality, we may assume that the image of the ramification divisor R by f in an apartment Σ containing an open set of La is defined by x2−x3 = 0, so that La ∩C for some chamber C in Σ is a line segment defined by x2 − x3 = ca, a constant. Hence R is defined by ω2 − ω3 = 0 on M1. We are done if f |e p(Ha) is non-singular, which implies that f ◦ p̃(Ha) is isometric to R. f ◦ p̃|Ha has singularity only along another ramification divisor R1 on M̃1, which is the stabilizer of an element ι ∈ W of order 2 since its image lies in the wall of a building. Hence we may assume that the image of R1 in Σ is defined by x1 − x2 = 0. As the local covering group generated by ι switches dx1 and dx2, we observe that f ◦ p̃|Ha is extended beyond f ◦ p̃|Ha ∩ f ◦ p̃(R1) in a unique way as a line segment in the adjacent chamber of C in Σ defined by x1 − x3 = ca for some constant ca determined by continuity. Let τ be the global one form on M1 annihilating R1, defined locally by 1/2(κ1 + κ2), where κi = (pf(dxi) ⊗ C)1,0. τ is the pull back of a holomorphic one form τo on E := A/ ker(τ) by αo : M1 α → A q → E. Let η = α∗ o)(τo). Fix zo ∈ Ha so that p̃(zo) is a regular point of f. Define Φ : M̃1 → R by Φ(z) = ∫ z zo η. We claim that for each apartment Σ for which La ∩ Σ is unbounded, there is a covering map Ψ : R = Φ(M̃1) → La ∩ Σ which is a local isometry, so that Ψ ◦ Φ(z) = f ◦ p̃(z). Suppose Φ(z1) = Φ(z2). Join zi to zo by a unique geodesic γi on Ha for i = 1, 2. It follows that for all t on γ1, there exists w(t) ∈ γ2 varying continuously with respect to t such that Φ(w(t)) = Φ(t) ∈ R. It suffices for us to show that f ◦ p̃(t) = f ◦ p̃(w(t)) for all t ∈ γ1 by continuity argument. This is clearly so for t in a small neighborhood of zo or a regular point of f from definition of τ .