Some things that can happen (with planar boundaries of...
Transcript of Some things that can happen (with planar boundaries of...
Some things that can happen (with planarboundaries of relatively hyperbolic groups)
Genevieve S. Walsh
Virginia 2016
(G ,P) relatively hyperbolic: Γ acts on X p.d. by isometries:
• X is a proper hyperbolic length space
• each x ∈ ∂X is either a conical limit point or a boundedparabolic point.
• The elements of P are exactly the maximal parabolic subgroupsof G (and f.g.)
P < G is parabolic if it is infinite, fixes some point xp and containsno loxodromics
Parabolic subgroup is bounded: (∂X \ xp)/P is compact.
Conical limit point: y ∈ ∂X : there exists (gi )i∈N and a, b ∈ ∂Xsuch that gi (y)→ a and gi (x)→ b for x ∈ ∂X \ y .
Relatively hyperbolic groups can be defined dynamically:
G acts on M (n.e., perfect, metrizable compact) as a convergencegroup (PD on ∂3(M))• action is uniform: G is hyperbolic. (Bowditch)• action is GF: (G ,P) is relatively hyperbolic where P is the set ofmaximal peripheral groups. (Yaman)
Global cut points can occur in relatively hyperbolic boundaries:
Global cut point =⇒ peripheral splitting, i.e., a splitting of thegroup over a subgroup of a peripheral subgroup. (Bowditch)
*boundaries of hyperbolic groups do not have global cut points*
Natural to Conjecture:If ∂(G ,P) is a Sierpinski carpet or S2, then G can be realized as aGF Kleinian group K . The maximal parabolic subgroups of K are asubset of P.
Recent progress: Cannon =⇒ Relative Cannon (Groves ManningSisto)
What about other planar relatively hyperbolic boundaries?
Groups can hide in global cut points: (Genus 2 surface group, fivewings, general group)“Tree-graded with respect to circles” - follows from Bowditch.
? Is it possible that:
∂(G ,P) connected planar without global cut points =⇒ G canbe realized as a Kleinian K (maximal parabolic subgroups of K area subset of P)
A Nice Class of Relatively hyperbolic boundaries: Schottky Sets(Bonk Kleiner Merenkov)
S: the complement of at least three round discs in S2.
Examples: Apollonian Gasket; Sierpinski carpet, Sierpinski carpetwith curves pinched.
Theorem: (HPW) K1, K2 Kleinian groups with Schottky set limitset. If (K1,P1) is Quasi-isometric to (K2,P2), then K1 and K2 arecommensurable in PSL(2,C).
Use BKM: If two such Schottky sets are quasi-symmetric, thenthey are conformal.
S = S2 \ ∪Di
The incidence graph is I (S) a vertex for every Di and an edgewhen Di ∩ Dj 6= ∅
Theorem * If S ' ∂(G ,P) and I (S) is connected and the incidencepoints are exactly the parabolic fixed points, G is virtually free.