Acylindrically hyperbolic group

Summarize Timeline Fact Check

Let G be a group with an isometric action on some geodesic hyperbolic metric space X. This action is called acylindrical^[1] if for every ${\displaystyle R\geq 0}$ there exist ${\displaystyle N>0,L>0}$ such that for every ${\displaystyle x,y\in X}$ with ${\displaystyle d(x,y)\geq L}$ one has

${\displaystyle \#\{g\in G\mid d(x,gx)\leq R,d(y,gy)\leq R\}\leq N.}$

If the above property holds for a specific ${\displaystyle R\geq 0}$, the action of G on X is called R-acylindrical. The notion of acylindricity provides a suitable substitute for being a proper action in the more general context where non-proper actions are allowed.

An acylindrical isometric action of a group G on a geodesic hyperbolic metric space X is non-elementary if G admits two independent hyperbolic isometries of X, that is, two loxodromic elements ${\displaystyle g,h\in G}$ such that their fixed point sets ${\displaystyle \{g^{+},g^{-}\}\subseteq \partial X}$ and ${\displaystyle \{h^{+},h^{-}\}\subseteq \partial X}$ are disjoint.

It is known (Theorem 1.1 in ^[1]) that an acylindrical action of a group G on a geodesic hyperbolic metric space X is non-elementary if and only if this action has unbounded orbits in X and the group G is not a finite extension of a cyclic group generated by loxodromic isometry of X.

A group G is called acylindrically hyperbolic if G admits a non-elementary acylindrical isometric action on some geodesic hyperbolic metric space X.

It is known (Theorem 1.2 in ^[1]) that for a group G the following conditions are equivalent:

• The group G is acylindrically hyperbolic.
• There exists a (possibly infinite) generating set S for G, such that the Cayley graph ${\displaystyle \Gamma (G,S)}$ is hyperbolic, and the natural translation action of G on ${\displaystyle \Gamma (G,S)}$ is a non-elementary acylindrical action.
• The group G is not virtually cyclic, and there exists an isometric action of G on a geodesic hyperbolic metric space X such that at least one element of G acts on X with the WPD ('Weakly Properly Discontinuous') property.
• The group G contains a proper infinite 'hyperbolically embedded' subgroup.^[2]

• Every acylindrically hyperbolic group G is SQ-universal, that is, every countable group embeds as a subgroup in some quotient group of G.
• The class of acylindrically hyperbolic groups is closed under taking infinite normal subgroups, and, more generally, under taking 's-normal' subgroups.^[1] Here a subgroup ${\displaystyle H\leq G}$ is called s-normal in ${\displaystyle G}$ if for every ${\displaystyle g\in G}$ one has ${\displaystyle |H\cap g^{-1}Hg|=\infty }$.
• If G is an acylindrically hyperbolic group and ${\displaystyle V=\mathbb {R} }$ or ${\displaystyle V=\ell ^{p}(G)}$ with ${\displaystyle p\in [1,\infty )}$ then the bounded cohomology ${\displaystyle H_{b}(G,V)}$ is infinite-dimensional.^[3][4][1]
• Every acylindrically hyperbolic group G admits a unique maximal normal finite subgroup denoted K(G).^[2]
• If G is an acylindrically hyperbolic group with K(G)={1} then G has infinite conjugacy classes of nontrivial elements, G is not inner amenable, and the reduced C*-algebra of G is simple with unique trace.^[2]
• There is a version of small cancellation theory over acylindrically hyperbolic groups, allowing one to produce many quotients of such groups with prescribed properties.^[5]
• Every finitely generated acylindrically hyperbolic group has cut points in all of its asymptotic cones.^[6]
• For a finitely generated acylindrically hyperbolic group G, the probability that the simple random walk on G of length n produces a 'generalized loxodromic element' in G converges to 1 exponentially fast as ${\displaystyle n\to \infty }$. ^[7]
• Every finitely generated acylindrically hyperbolic group G has exponential conjugacy growth, meaning that the number of distinct conjugacy classes of elements of G coming from the ball of radius n in the Cayley graph of G grows exponentially in n. ^[8]