Free subgroups of the homeomorphism group of the circle
By Gregory Margulis. 2000
85 citations recorded by the source corpus.
Abstract Tits alternative says that a finitely generated linear group either contains a non-commutative free subgroup or is virtually solvable. It is known ( see [2]) that an analogue of Tits alternative is not true for subgroups of the group Homeo(S 1 ) of all homeomorphisms of the circle S 1 and even for subgroups of the group of C ∞ -diffeomorphisms of S 1 . The main purpose of this Note is to prove a conjecture of E. Ghys which can be viewed as a replacement of Tits alternative for Homeo(S 1 ) and which says that if G is a subgroup of Homeo(S 1 ) containing no free non-commutative subgroup then there is a G -invariant probability measure on S 1 .
Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/0ee1d4325fdba3d3f473e43581bc47886cb23040
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Citations in the corpus
- Groups acting on manifolds: around the Zimmer program (2008, background): “M. Then either Γ contains a free group on two generators or Γ preserves some measure on M. The best evidence for this conjecture to date is a theorem of Margulis which proves the conjecture for M= S1 [160]. Ghys has also asked if the more exact analogue of Tits’ theorem might be true for analytic di”
- GROUP ACTIONS ON 1-MANIFOLDS: A LIST OF VERY CONCRETE OPEN QUESTIONS (2017, background): “On the other hand, as explained before, an action of type (ii) factors through a locally contracting action on the circle, which implies the presence of a free subgroup by a theorem of Margulis [93] (see also [53]).”