Some things that can happen (with planarboundaries of relatively hyperbolic groups)

Genevieve S. Walsh

Virginia 2016

Joint with various people: Francois Dahmani, Chris Hruska, LuisaPaoluzzi.

Goal: understand geometrically finite groups with planar boundary.

Example: The limit set of a geometrically finite Kleinian group:

(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.)

M = ∂X = ∂Bow (G ,P) is well-defined for (G ,P) (Bowditch)

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)

Some relatively hyperbolic boundaries: S1,

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*

What kind of (G ,P) can occur for planar relatively hyperbolicgroups?

Dahmani: If ∂(G ,P) is a Sierpinski carpet or S2, every element ofP is a virtual surface group.

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.

Free groups as peripheral groups

? 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)

Panic now about hidden cut points

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.

a : z → z + 1 b : z → 1/(2iz + 1)(fixed point of ABab is (-1 +i)/2 )

A Kleinian group with boundary (homeomorphic to) A iscommensurable to this group.

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.

Generalized from Sierpinski carpets:

Theorem * S = S2 \ ∪Di . The ∂Di are exactly the non-separatingcircles in S.