TY - JOUR

T1 - Discrete subgroups of the special linear group with thin limit sets

AU - Yun, Aaram

N1 - Publisher Copyright:
© 2016 American Mathematical Society.

PY - 2017

Y1 - 2017

N2 - In this paper, we construct a discrete Zariski-dense subgroup Γ of SL(n+1,R) whose limit set on Pn is ‘thin’, that is, contained in a CN-smooth curve, for any n ≥ 3 and N > 0. We achieve this by applying the ping-pong lemma to the action of a specially chosen generating set S on the N-th order jet bundle over Pn. We also show that in a sense this is the best possible result: we show that there does not exist any Zariski-dense subgroup Γ ⊆ SL(3,R) whose limit set is contained in a C2-smooth curve, and there does not exist any Zariski-dense subgroup Γ ⊆ SL(n+1,R) whose limit set is contained in a C∞-smooth curve.

AB - In this paper, we construct a discrete Zariski-dense subgroup Γ of SL(n+1,R) whose limit set on Pn is ‘thin’, that is, contained in a CN-smooth curve, for any n ≥ 3 and N > 0. We achieve this by applying the ping-pong lemma to the action of a specially chosen generating set S on the N-th order jet bundle over Pn. We also show that in a sense this is the best possible result: we show that there does not exist any Zariski-dense subgroup Γ ⊆ SL(3,R) whose limit set is contained in a C2-smooth curve, and there does not exist any Zariski-dense subgroup Γ ⊆ SL(n+1,R) whose limit set is contained in a C∞-smooth curve.

UR - http://www.scopus.com/inward/record.url?scp=84992047540&partnerID=8YFLogxK

U2 - 10.1090/tran/6753

DO - 10.1090/tran/6753

M3 - Article

AN - SCOPUS:84992047540

VL - 369

SP - 365

EP - 407

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 1

ER -