TY - JOUR
T1 - Manifolds counting and class field towers
AU - Belolipetsky, Mikhail
AU - Lubotzky, Alexander
PY - 2012/4/1
Y1 - 2012/4/1
N2 - In Burger et al. (2002) [12] and Goldfeld et al. (2004) [17] it was conjectured that if H is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in H of covolume at most x is x (γ(H)+o(1))logx/loglogx where γ(H) is an explicit constant computable from the (absolute) root system of H. In this paper we prove that this conjecture is false. In fact, we show that the growth is at rate x clogx. A crucial ingredient of the proof is the existence of towers of field extensions with bounded root discriminant which follows from the seminal work of Golod and Shafarevich on class field towers.
AB - In Burger et al. (2002) [12] and Goldfeld et al. (2004) [17] it was conjectured that if H is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in H of covolume at most x is x (γ(H)+o(1))logx/loglogx where γ(H) is an explicit constant computable from the (absolute) root system of H. In this paper we prove that this conjecture is false. In fact, we show that the growth is at rate x clogx. A crucial ingredient of the proof is the existence of towers of field extensions with bounded root discriminant which follows from the seminal work of Golod and Shafarevich on class field towers.
KW - Arithmetic subgroups
KW - Class field towers
KW - Counting lattices
KW - Lattices in higher rank Lie groups
KW - Subgroup growth
UR - http://www.scopus.com/inward/record.url?scp=84857279471&partnerID=8YFLogxK
U2 - 10.1016/j.aim.2012.02.002
DO - 10.1016/j.aim.2012.02.002
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AN - SCOPUS:84857279471
SN - 0001-8708
VL - 229
SP - 3123
EP - 3146
JO - Advances in Mathematics
JF - Advances in Mathematics
IS - 6
ER -