Non-concentration of primes in Γ\PSL2(ℝ)

Abstract This paper generalizes the result of Sarnak and Ubis [9] about non-concentration of primes in horocycle orbits on PSL 2 (ℤ)PSL 2 (ℝ) to any lattice in PSL 2 (ℝ). The proof combines the asymptotic result of Strömbergsson [11] and Venkatesh’s method [12] with the approach of Sarnak and Ubis of approximating horocycle pieces with periodic horocycles. The key step is to establish a dichotomy between { ξh ( t ), t ∈ [0, T ]} having good equidistribution in ΓPSL 2 (ℝ) and it being approximable by closed horocycle pieces with small period. In a follow-up paper, a similar approach will be used to show equidistribution of ξh ( n 1+ γ ) for small γ > 0, generalizing Venkatesh’s result [12] to non-compact Γ.