(1) Institute of Mathematics, Hebrew University, 91904 Jerusalem, Israel;(2) Institute of Mathematics, Hebrew University, 91904 Jerusalem, Israel;(3) School of Mathematics, Tata Institute of Fundamental Research, 400-005 Bombay, India
Abstract:
Let G be a semisimple Lie group of rank ⩾2 and Γ an irreducible lattice. Γ has two natural metrics: a metric inherited from
a Riemannian metric on the ambient Lie group and a word metric defined with respect to some finite set of generators. Confirming
a conjecture of D. Kazhdan (cf. Gromov Gr2]) we show that these metrics are Lipschitz equivalent. It is shown that a cyclic
subgroup of Γ is virtually unipotent if and only if it has exponential growth with respect to the generators of Γ.