TY - JOUR
T1 - DIVISIBILITY PROPERTIES OF HIGHER RANK LATTICES
AU - Einsiedler, Manfred
AU - Mozes, Shahar
N1 - Publisher Copyright:
© 2016, Springer Science+Business Media New York.
PY - 2016/12/1
Y1 - 2016/12/1
N2 - We discuss a relationship between the dynamical properties of a maximal diagonalizable group A on certain arithmetic quotients and arithmetic properties of the lattice. In particular, we consider the semigroup of all integer quaternions under multiplication. For this semigroup we use measure rigidity theorems to prove that the set of elements that are not divisible by a given reduced quaternion is very small: We show that any quaternion that has a sufficiently divisible norm is also divisible by the given quaternion. Restricting to the quaternions that have norm equal to products of powers of primes from a given list (containing at least two) we show that the set of exceptions has subexponential growth.
AB - We discuss a relationship between the dynamical properties of a maximal diagonalizable group A on certain arithmetic quotients and arithmetic properties of the lattice. In particular, we consider the semigroup of all integer quaternions under multiplication. For this semigroup we use measure rigidity theorems to prove that the set of elements that are not divisible by a given reduced quaternion is very small: We show that any quaternion that has a sufficiently divisible norm is also divisible by the given quaternion. Restricting to the quaternions that have norm equal to products of powers of primes from a given list (containing at least two) we show that the set of exceptions has subexponential growth.
UR - http://www.scopus.com/inward/record.url?scp=84979209190&partnerID=8YFLogxK
U2 - 10.1007/s00031-016-9399-0
DO - 10.1007/s00031-016-9399-0
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:84979209190
SN - 1083-4362
VL - 21
SP - 1039
EP - 1062
JO - Transformation Groups
JF - Transformation Groups
IS - 4
ER -