Hyperbolic group

Hyperbolic groups (also: word-hyperbolic groups , Gromov hyperbolic groups , negatively curved groups ) are one of the central themes of geometric group theory .

The term was introduced by Michail Leonidowitsch Gromow in the 1980s, but the use of geometric methods in group theory has a tradition that goes back to Max Dehn's use of hyperbolic geometry to solve the word problem for fundamental groups of compact surfaces . In a sense, almost all groups are hyperbolic. Numerous methods from the geometry of negatively curved spaces can be transferred to hyperbolic groups and thus made usable for group theory.

definition

A finitely generated group is hyperbolic if the Cayley graph assigned to a finite generating system - is hyperbolic for a . This definition is independent of the choice of the finite generating system. ${\displaystyle \delta }$ ${\displaystyle \delta >0}$

In more detail:

The Cayley graph of the free group with two generators and ${\displaystyle a}$ ${\displaystyle b}$

The Cayley graph assigned to a finite generating system S of a group G is the graph defined as follows : The node set is the group , the edge set consists of pairs of the form , with any group element and one element being off . The picture on the right shows the Cayley graph of the free group created by two elements . ${\displaystyle (V,E)}$ ${\displaystyle V}$ ${\displaystyle G}$ ${\displaystyle E}$ ${\displaystyle (g,gs)}$ ${\displaystyle g}$ ${\displaystyle s}$ ${\displaystyle S\cup S^{-1}}$ ${\displaystyle S=\left\{a,b\right\}}$

By specifying that all edges are length , the Cayley graph becomes a metric space. (The induced metric on the node set is called the word metric of the group .) ${\displaystyle 1}$ ${\displaystyle G}$ ${\displaystyle G}$

Quasi-isometric Cayley graphs are obtained for various finite generating systems. All geometric properties of graphs, which are determined except for quasi-isometry, therefore correspond to properties of groups.

A geodesic triangle in a negatively curved surface

A δ-thin triangle

A metric space is called -hyperbolic for one if all geodesic triangles are δ-thin , i.e. H. every edge of the triangle is contained in the neighborhood of the union of the other two edges: ${\displaystyle \delta }$ ${\displaystyle \delta >0}$ ${\displaystyle \delta }$

${\displaystyle [x,y]\subseteq B_{\delta }([y,z]\cup [z,x]),}$

${\displaystyle [y,z]\subseteq B_{\delta }([z,x]\cup [x,y]),}$

${\displaystyle [z,x]\subseteq B_{\delta }([x,y]\cup [y,z]).}$

This condition is fulfilled, for example, for geodetic triangles in trees with or in the hyperbolic plane , more generally for geodetic triangles in simply connected Riemannian manifolds with negative sectional curvature . In Euclidean space, on the other hand, the property is not fulfilled: for each one can get to a triangle in a triangle by simple scaling with a constant positive, dependent factor, in which the -surrounding of two edges is not the -surrounding of the third edge in the triangle includes. ${\displaystyle \delta =0}$ ${\displaystyle \delta =\ln \left({\sqrt {2}}+1\right)}$ ${\displaystyle \delta >0}$ ${\displaystyle \delta }$ ${\displaystyle \delta }$ ${\displaystyle \delta }$

If two metric spaces and are quasi-isometric, then -hyperbolic is for one and only if -hyperbolic is for one (possibly different) . In particular, the Cayley graph assigned to a finite generating system is a group -hyperbolic for a if and only if this applies to every finite generating system. ${\displaystyle (X_{1},d_{1})}$ ${\displaystyle (X_{2},d_{2})}$ ${\displaystyle (X_{1},d_{1})}$ ${\displaystyle \delta }$ ${\displaystyle \delta >0}$ ${\displaystyle (X_{2},d_{2})}$ ${\displaystyle \delta ^{\prime }}$ ${\displaystyle \delta ^{\prime }>0}$ ${\displaystyle \delta }$ ${\displaystyle \delta >0}$

With this one can then define independently of the chosen finite generating system of a group : the group is hyperbolic if the Cayley graph is hyperbolic for one . ${\displaystyle S}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle \delta }$ ${\displaystyle \delta >0}$

Examples

• Finite groups and virtual cyclic groups are hyperbolic, these groups are often referred to as elementary hyperbolic groups.
• Free groups that are finally generated are hyperbolic.
• Fundamental groups of compact Riemannian manifolds with negative section curvature are hyperbolic. This includes in particular fundamental groups of compact hyperbolic manifolds , for example fundamental groups of compact surfaces with negative Euler characteristics .
• A “randomly chosen” group is hyperbolic. That means more precisely: For a (arbitrary, but fixed) natural number and a with, consider all groups with generators and (at most) relations of length (at most) for every natural number . Let be the proportion of hyperbolic groups in this set of groups. Gromov has proven that for towards infinity the share goes towards 100%. ${\displaystyle n}$ ${\displaystyle d\in \mathbb {R} }$ ${\displaystyle 1/2<d<1}$ ${\displaystyle L}$ ${\displaystyle n}$ ${\displaystyle dL}$ ${\displaystyle L}$ ${\displaystyle P(n,d,L)}$ ${\displaystyle L}$ ${\displaystyle P(n,d,L)}$
• A group that contains as a subgroup is not hyperbolic. ${\displaystyle \mathbb {Z} \oplus \mathbb {Z} }$

Applications

Various conjectures that can be formulated for any group (and generally open) have been proven for the class of hyperbolic groups using their special geometry. This includes:

• the Novikov conjecture
• the Baum-Connes conjecture
• the Farrell-Jones Conjecture

Edge at infinity

${\displaystyle \delta }$-hyperbolische rooms have mostly as Gromov-edge designated edge at infinity . This is defined as the set of equivalence classes of geodetic rays, where two rays are equivalent if and only if they are finite apart. ${\displaystyle X}$ ${\displaystyle \partial X}$

After selection of a fixed base point to define the topology of as follows: As a basis for neighborhoods of a point using all with , wherein the set of all , so that and through of outgoing geodetic rays are represented for which is. Here denotes the Gromov product . The topology on is independent of the one selected . ${\displaystyle x\in X}$ ${\displaystyle \partial X}$ ${\displaystyle p\in \partial X}$ ${\displaystyle V(p,r)\subset \partial X}$ ${\displaystyle r\geq 0}$ ${\displaystyle V(p,r)}$ ${\displaystyle q\in \partial X}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle x}$ ${\displaystyle \gamma _{1},\gamma _{2}}$ ${\displaystyle \liminf _{t\rightarrow \infty }(\gamma _{1}(t),\gamma _{2}(t))_{x}\geq r}$ ${\displaystyle (.,.)_{x}}$ ${\displaystyle (y,z)_{x}={\frac {1}{2}}(d(x,y)+d(x,z)-d(y,z))}$ ${\displaystyle \partial X}$ ${\displaystyle x}$

Quasi-isometric spaces have homeomorphic boundaries at infinity. In particular, the boundary of a hyperbolic group is well-defined (independent of the generating system ) as the boundary at infinity of the Cayley graph. Examples: for free groups the boundary at infinity is a Cantor set , for fundamental groups of compact -dimensional Riemannian manifolds with negative section curvature the boundary at infinity is a -dimensional sphere , for "most" hyperbolic groups the boundary at infinity is a Menger sponge . ${\displaystyle S}$ ${\displaystyle n}$ ${\displaystyle (n-1)}$

Quasi-isometric, especially isometrics, a -hyperbolischen space act as homeomorphisms on . In particular, every hyperbolic group acts through isometries on its Cayley graph and thus through homeomorphisms on its boundary at infinity. The action of the hyperbolic group on the edge at infinity is a “chaotic” dynamic system . ${\displaystyle \delta }$ ${\displaystyle X}$ ${\displaystyle \partial X}$ ${\displaystyle G}$

A hyperbolic group acts as a convergence group on its edge at infinity and this allows a topological characterization of hyperbolic groups: A group is hyperbolic if and only if it acts as a uniform convergence group on a perfect, compact, metrizable space.

See also

• Relatively hyperbolic group
• Acylindrical hyperbolic group

literature

• Mikhail Gromov: Hyperbolic groups. In: Stephen M. Gersten (Ed.): Essays in group theory (= Mathematical Sciences Research Institute Publications. Vol. 8). Springer, New York NY et al. 1987, pp. 75-263, ISBN 0-387-96618-8 . online (pdf)
• Michel Coornaert, Thomas Delzant, Athanase Papadopoulos: Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov (= Lecture Notes in Mathematics. Vol. 1441). Springer, Berlin et al. 1990, ISBN 3-540-52977-2 .
• Étienne Ghys, Pierre de la Harpe (ed.): Sur les groupes hyperboliques d'après Mikhael Gromov (= Progress in Mathematics. Vol. 83). Birkhäuser Boston, Inc., Boston MA et al. 1990, ISBN 0-8176-3508-4 .

Individual evidence

1. ^ Mikhail Gromov: Random walk in random groups. In: Geometric & Functional Analysis. Vol. 13, No. 1, 2003, pp. 73–146, doi : 10.1007 / s000390300002 , digitized version  ( page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice. (PDF; 787.25 kB).  
2. ^ Alain Connes , Henri Moscovici : Cyclic cohomology, the Novikov conjecture and hyperbolic groups. In: Topology. Vol. 29, No. 3, 1990, pp. 345-388, doi : 10.1016 / 0040-9383 (90) 90003-3 .
3. ↑ Igor Mineyev, Guoliang Yu: The Baum-Connes conjecture for hyperbolic groups. In: Inventiones Mathematicae. Vol. 149, No. 1, 2002, pp. 97-122, doi : 10.1007 / s002220200214 , digitized version (PDF; 250 kB).
4. ^ Arthur Bartels, Wolfgang Lück , Holger Reich: The K-theoretic Farrell-Jones conjecture for hyperbolic groups. In: Inventiones Mathematicae. Vol. 172, No. 1, 2008, pp. 29-70, doi : 10.1007 / s00222-007-0093-7 , digital version (PDF; 470.9 kB).
5. ↑ Ilya Kapovich, Nadia Benakli: Boundaries of hyperbolic groups. In: Sean Cleary, Robert Gilman, Alexei G. Myasnikov, Vladimir Shpilrain (Eds.): Combinatorial and geometric group theory. AMS Special Session Combinatorial Group Theory, Nov 4-5, 2000, New York, New York. AMS Special Session Computational Group Theory, April 28-29, 2001, Hoboken, New Jersey (= Contemporary Mathematics. Vol. 296). American Mathematical Society, Providence RI 2002, ISBN 0-8218-2822-3 , pp. 39-93, digitized version (PDF; 488 kB).