Distribution of holonomy about closed geodesics in a product of hyperbolic planes

Abstract

Abstract. Let M = Γ\H (n) , where H (n) is a product of n + 1 hyperbolic planes and Γ ⊂ PSL(2, R)n+1 is an irreducible cocompact lattice. We consider closed geodesics on M that propagate locally only in one factor. We show that, as the length tends to infinity, the holonomy rotations attached to these geodesics become equidistributed in PSO (2) n with respect to a certain measure. For the special case of lattices derived from quaternion algebras, we can give another interpretation of the holonomy angles under which this measure arises naturally.

Hybrid trace formula for a non-uniform irreducible lattice inPSL2(R)n

“…The results on counting elliptic-hyperbolic classes and on the distribution of their elliptic parts follow from Theorem 3 exactly as in [Ke10]. Specifically, we note that the family of functions Using the same argument as in the proof of [Ke10, Proposition 3.3], and noting that for congruence groups we have the bound λ 0 (m) ≥ 1 4 −( 7 64 ) 2 for the spectral gap, we get…”

“…For Γ ⊂ PSL 2 (R) n an irreducible uniform torsion free lattice, the trace formula for these hybrid forms takes the following form (see [Ke10,Theorem 7']). where h(r) is any even holomorphic function with Fourier transformĥ compactly supported, λ k (m) = 1 4 + r 2 k,m , and |m| * = n−1 j=1 (2|m j | − 1).…”

“…For Γ ⊂ PSL 2 (R) n an irreducible uniform torsion free lattice, the trace formula for these hybrid forms takes the following form (see [Ke10,Theorem 7']).…”

A Uniform Strong Spectral Gap for Congruence Covers of a Compact Quotient of

“…There is an alternative way to obtain this formula which is applicable also when some |m j | = 1. We note that if all |m j | = 1, there is an additional term of (−1) d−d0 h(i/2) entering on the spectral side (see [Ke,Se2]).…”

“…Proof. We recall the hybrid trace formula (see [Ke,Theorem 7']), where the notation ′ indicates that we are summing over conjugacy classes of elements satisfying |tr(γ 2 )| < 2.…”