Relatively hyperbolic group

In mathematics , relatively hyperbolic groups are a concept of geometric group theory that generalizes the concept of the hyperbolic group and in particular includes the fundamental groups of hyperbolic manifolds of finite volume, while only the fundamental groups of compact hyperbolic manifolds are hyperbolic groups.

The relative hyperbolicity of a group is defined relative to a family of subgroups. One also speaks of relatively hyperbolic groups as groups that are hyperbolic relative to a real subgroup.

definition

Let be a finitely generated group and a finite set of conjugation classes of subsets of . ${\displaystyle G}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle G}$

${\displaystyle (G,{\mathcal {P}})}$is relatively hyperbolic if there is an actually discontinuous group action of by isometrics on an actual hyperbolic space such that ${\displaystyle G}$ ${\displaystyle X}$

• every point of the boundary at infinity is either a conical limit point or a bounded parabolic fixed point , and ${\displaystyle \partial _{\infty }X}$
• the subgroups in the conjugation classes from are exactly the maximum parabolic subgroups of . ${\displaystyle {\mathcal {P}}}$ ${\displaystyle G}$

While the space is not unambiguously determined apart from quasi-isometry , the boundary at infinity is clearly defined and is called the boundary at infinity of the relatively hyperbolic group. ${\displaystyle X}$ ${\displaystyle \partial _{\infty }X}$ ${\displaystyle \partial G}$

A subgroup is also said to be relatively hyperbolic or hyperbolic relative to when the set of subgroups to be conjugated is relatively hyperbolic. Analogously, for a finite set of subgroups , one says that hyperbolic is relative to if it is hyperbolic for the set of conjugated subgroups . ${\displaystyle P\subset G}$ ${\displaystyle (G,P)}$ ${\displaystyle G}$ ${\displaystyle P}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle P}$ ${\displaystyle (G,{\mathcal {P}})}$ ${\displaystyle P_{1},\ldots ,P_{n}\subset G}$ ${\displaystyle G}$ ${\displaystyle P_{1},\ldots ,P_{n}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle P_{1},\ldots ,P_{n}}$ ${\displaystyle (G,{\mathcal {P}})}$

Equivalent Definitions

Bowditch's definition

A group acts on a fine , hyperbolic graph with finite edge stabilizers and finitely many orbites of edges. be the set of stabilizers of nodes of infinite valence. Then the pair is relatively hyperbolic. ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle (G,{\mathcal {P}})}$

Color definition

For a finitely generated group and a finite set of conjugation classes of subgroups of, let the graph whose nodes are the nodes of the Cayley graph and for each and its edges those of (with length 1) and those between (with length 1/2). is relatively hyperbolic, when this graph is hyperbolic, and if limited cosets penetration (bounded coset Penetration, BCP), d is. H. So that if two - Quasigeodäten without backtracking with and are, then: ${\displaystyle G}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle G}$ ${\displaystyle {\mathcal {C}}(G,{\mathcal {P}},A)}$ ${\displaystyle {\mathcal {C}}(G,A)}$ ${\displaystyle v_{j}P}$ ${\displaystyle P\in {\mathcal {P}}}$ ${\displaystyle {\mathcal {C}}(G,A)}$ ${\displaystyle (G,{\mathcal {P}})}$ ${\displaystyle \forall \lambda >1\exists a>0}$ ${\displaystyle \gamma _{1},\gamma _{2}}$ ${\displaystyle (\lambda ,0)}$ ${\displaystyle \gamma _{1}(0)=\gamma _{2}(0)}$ ${\displaystyle d(\gamma _{1}(1),\gamma _{2}(1))\leq 1}$

• if one penetrates, but not, then the distance between the input and output nodes is at most , ${\displaystyle \gamma _{1}}$ ${\displaystyle P\in {\mathcal {P}}}$ ${\displaystyle \gamma _{2}}$ ${\displaystyle \gamma _{1}}$ ${\displaystyle a}$
• if both penetrate one, then the distance of the input nodes from and is at most and the distance of the output nodes from and is at most . ${\displaystyle \gamma _{1},\gamma _{2}}$ ${\displaystyle P\in {\mathcal {P}}}$ ${\displaystyle \gamma _{1}}$ ${\displaystyle \gamma _{2}}$ ${\displaystyle a}$ ${\displaystyle \gamma _{1}}$ ${\displaystyle \gamma _{2}}$ ${\displaystyle a}$

The edge at infinity is then the union . ${\displaystyle \partial {\mathcal {C}}(G,{\mathcal {P}},A)\cup V_{P}}$

Special cases

• Let be a hyperbolic surface with a connected , totally geodetic edge . Then the fundamental group is a free group . The pair is relatively hyperbolic and its edge at infinity is a Cantor set . If the homotopy class is the boundary curve and the cyclic subgroup it generates is, then the pair is also a relatively hyperbolic group for its conjugation class , whose edge at infinity is a circle (the edge at infinity of the hyperbolic plane ). ${\displaystyle S}$ ${\displaystyle \pi _{1}S}$ ${\displaystyle F}$ ${\displaystyle (F,\emptyset )}$ ${\displaystyle c}$ ${\displaystyle \langle c\rangle }$ ${\displaystyle F}$ ${\displaystyle \left[\langle c\rangle \right]}$ ${\displaystyle (F,\left[\langle c\rangle \right])}$
• Let be a CAT (0) group with isolated flax and consist of the (conjugation classes of the) stabilizers of flax. Then the pair is relatively hyperbolic and its edge at infinity arises from the collapse of the edges at infinity of the flax to one point each. Let, for example, be the fundamental group of a hyperbolic manifold of finite volume , then it is a CAT (0) group and its edge at infinity is a Sierpinski carpet , in the universal superposition horospheres form a family of isolated flax and the pair defined in this way has as edge a sphere in the infinite . ${\displaystyle G}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle (G,{\mathcal {P}})}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle (G,{\mathcal {P}})}$
• Let be a hyperbolic group and an almost malnormal family of quasi-convex subgroups , then a relatively hyperbolic group, whose boundary at infinity is obtained from that of by collapsing the boundaries at infinity of the subgroups in the conjugation classes of . Let, for example, be the fundamental group of a hyperbolic manifold with a totally geodesic boundary and consist of the conjugation classes of the fundamental groups of the boundary components, then it is relatively hyperbolic and the boundary at infinity is a sphere. ${\displaystyle G}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle (G,{\mathcal {P}})}$ ${\displaystyle G}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle G}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle (G,{\mathcal {P}})}$

Examples

• For a hyperbolic group is relatively hyperbolic. ${\displaystyle G}$ ${\displaystyle (G,\emptyset )}$
• Let be a compact manifold with a connected boundary . If the interior of has a hyperbolic metric of finite volume, then it is relatively hyperbolic. ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle (\pi _{1}M,\pi _{1}\partial M)}$
• The couple are not relatively hyperbolic. ${\displaystyle (\mathbb {Z} \oplus \mathbb {Z} ,\mathbb {Z} \oplus 1)}$
• The mapping class group of a surface by gender is not relatively hyperbolic to any real subgroup. ${\displaystyle g\geq 2}$
• The outer automorphism group of a free group of rank is not relatively hyperbolic to any true subgroup. ${\displaystyle n\geq 3}$
• If is relative hyperbolic and hyperbolic, then is hyperbolic. ${\displaystyle (G,H)}$ ${\displaystyle H}$ ${\displaystyle G}$

literature

• B. Bowditch : Relatively hyperbolic groups , Int. J. Alg. Comp. 22 (2012)
• Benson Farb : Relatively hyperbolic groups , Geom. Funct. Anal. 8: 810-840 (1998).
• Daniel Groves , Jason Manning : Dehn filling in relatively hyperbolic groups , Isr. J. of Math. 168 (2008), 317-429.

Individual evidence

1. ↑ C. Hruska , B. Kleiner : Hadamard spaces with isolated flats, with an appendix written jointly with Mohamad Hindawi , Geom. Topol. 9: 1501-1538 (2005)
2. ↑ J. Manning , O. Wang: Cohomology of the Bowditch boundary , Preprint (2018)