TL;DR: In this article, the integral Novikov conjecture for arithmetic groups is proved using the controlled K-theory associated with the compactifications of symmetric spaces of non-compact type X.
TL;DR: In this article, the genera of arithmetic groups ⟨Γ1(N), Φ⟩ for N ≥ 1 and Φ the Fricke involution were derived.
Abstract: We find the genera of arithmetic groups ⟨Γ1(N), Φ⟩ for N ≥ 1 and Φ the Fricke involution, from which we derive that the genus g(⟨Γ1(N), Φ⟩) is zero if and only if 1 ≤ N ≤ 12 and N = 14, 15.