TY - JOUR

T1 - Margulis lemma for compact lie groups

AU - Mazur, Marcin

AU - Rong, Xiaochun

AU - Wang, Yusheng

N1 - Funding Information:
X. Rong: supported partially by NSF Grant DMS 0504534 and by a reach found from Beijing Normal University.
Funding Information:
Y. Wang: supported partially by LMAM of Peking University and by NSFC 10671018.

PY - 2008/2

Y1 - 2008/2

N2 - We improve Margulis lemma for a compact connected Lie group G: there is a neighborhood U of the identity such that for any finite subgroup Γ ⊂ G, U ∩ Γ generates an abelian group. We show that for each n, there exists an integer w(n) > 0, such that if H is a closed subgroup of a compact connected Lie group G of dimension n, then the quotient group, H/H 0, has an abelian subgroup of index ≤ w(n) , where H 0 is the identity component of H. As an application, we show that the fundamental group of the homogeneous space G/H has an abelian subgroup of index ≤ w(n). We show this same property for the fundamental groups of almost non-negatively curved n-manifolds whose universal coverings are not collapsed.

AB - We improve Margulis lemma for a compact connected Lie group G: there is a neighborhood U of the identity such that for any finite subgroup Γ ⊂ G, U ∩ Γ generates an abelian group. We show that for each n, there exists an integer w(n) > 0, such that if H is a closed subgroup of a compact connected Lie group G of dimension n, then the quotient group, H/H 0, has an abelian subgroup of index ≤ w(n) , where H 0 is the identity component of H. As an application, we show that the fundamental group of the homogeneous space G/H has an abelian subgroup of index ≤ w(n). We show this same property for the fundamental groups of almost non-negatively curved n-manifolds whose universal coverings are not collapsed.

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U2 - 10.1007/s00209-007-0178-4

DO - 10.1007/s00209-007-0178-4

M3 - Article

AN - SCOPUS:36448967821

SN - 0025-5874

VL - 258

SP - 395

EP - 406

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 2

ER -