TY - JOUR

T1 - Anosov automorphisms on compact nilmanifolds associated with graphs

AU - Dani, S. G.

AU - Mainkar, Meera G.

PY - 2005/6

Y1 - 2005/6

N2 - We associate with each graph (S, E) a 2-step simply connected nilpotent Lie group N and a lattice Γ in N. We determine the group of Lie automorphisms of N and apply the result to describe a necessary and sufficient condition, in terms of the graph, for the compact nilmanifold N/Γ to admit an Anosov automorphism. Using the criterion we obtain new examples of compact nilmanifolds admitting Anosov automorphisms, and conclude that for every n ≥ 17 there exist a n-dimensional 2-step simply connected nilpotent Lie group N which is indecomposable (not a direct product of lower dimensional nilpotent Lie groups), and a lattice Γ in N such that N/Γ admits an Anosov automorphism; we give also a lower bound on the number of mutually nonisomorphic Lie groups N of a given dimension, satisfying the condition. Necessary and sufficient conditions are also described for a compact nilmanifold as above to admit ergodic automorphisms.

AB - We associate with each graph (S, E) a 2-step simply connected nilpotent Lie group N and a lattice Γ in N. We determine the group of Lie automorphisms of N and apply the result to describe a necessary and sufficient condition, in terms of the graph, for the compact nilmanifold N/Γ to admit an Anosov automorphism. Using the criterion we obtain new examples of compact nilmanifolds admitting Anosov automorphisms, and conclude that for every n ≥ 17 there exist a n-dimensional 2-step simply connected nilpotent Lie group N which is indecomposable (not a direct product of lower dimensional nilpotent Lie groups), and a lattice Γ in N such that N/Γ admits an Anosov automorphism; we give also a lower bound on the number of mutually nonisomorphic Lie groups N of a given dimension, satisfying the condition. Necessary and sufficient conditions are also described for a compact nilmanifold as above to admit ergodic automorphisms.

UR - http://www.scopus.com/inward/record.url?scp=20144376367&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-04-03518-4

DO - 10.1090/S0002-9947-04-03518-4

M3 - Article

AN - SCOPUS:20144376367

VL - 357

SP - 2235

EP - 2251

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 6

ER -