Acylindrical hyperbolic group

Acylindrical hyperbolic groups are a term used in geometric group theory .

They form a large class of groups with "hyperbolic properties" to which, in addition to hyperbolic groups, for example, mapping class groups and groups of external automorphisms also belong. For acylindrical hyperbolic groups numerous of the "largeness properties" of free and hyperbolic groups apply.

Acylindrical effects

An action of a group on a metric space is called acylindrical if there are positive numbers for every positive number , so that for all with ${\displaystyle G}$ ${\displaystyle S}$ ${\displaystyle \epsilon }$ ${\displaystyle R,N}$ ${\displaystyle x,y\in S}$

${\displaystyle d(x,y)>R}$

at most group elements with ${\displaystyle N}$ ${\displaystyle g\in G}$

${\displaystyle d(x,gx)\leq \epsilon }$ and ${\displaystyle d(y,gy)\leq \epsilon }$

exist.

Acylindrical hyperbolic groups

A group is called acylindrical hyperbolic if it fulfills one of the following equivalent conditions:

- there is a Gromov hyperbolic space in which it appears non-elementarily acylindrical,

- there is a (possibly infinite) generating system , so that the Cayley graph is hyperbolic and has more than two boundary points and the natural action of the group on the Cayley graph is acylindrical,

- the group is not virtually cyclic and acts on a Gromov hyperbolic space, so that at least one group element acts as a loxodromic isometry and fulfills the WPD condition ,

- the group contains an infinite, hyperbolically embedded , real subgroup

Examples

The following classes of groups are acylindrical hyperbolic:

- non-elementary hyperbolic groups ,

- not virtually cyclical, relatively hyperbolic groups with real peripheral subgroups,

- the mapping class groups of closed areas by gender , ${\displaystyle g\geq 1}$

- the group of external automorphisms of a free group of rank , ${\displaystyle n\geq 2}$

- not virtually cyclical groups that actually work on a CAT (0) space .

properties

Let be an acylindrical hyperbolic group that appears non-elementarily acylindrical in a Gromov hyperbolic space . Then: ${\displaystyle G}$ ${\displaystyle X}$

• every countable group can be embedded in a factor group of . ${\displaystyle Q}$ ${\displaystyle G}$

• there is one that acts as a loxodromic isometry on . ${\displaystyle g\in G}$ ${\displaystyle X}$

• ${\displaystyle G}$contains a free subgroup whose orbites are embedded in quasi-isometric . ${\displaystyle X}$

• Bounded cohomology in degrees 2 and 3 is infinite-dimensional.

literature

• D. Osin, Acylindrical hyperbolic groups, Trans. Amer. Math. Soc. 368 (2016), no. 2, 851-888.

Web links

• T. Koberda: WHAT IS ... an acylindrical group action? (Notices of the AMS, January 2018)

Individual evidence

1. ↑ Mladen Bestvina, Koji Fujiwara, Bounded cohomology of subgroups of mapping class groups, Geom. Topol. 6: 69-89 (2002)
2. ↑ F. Dahmani, V. Guirardel, D. Osin, Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces (2011), arXiv: 1111.7048