Gromov hyperbolic space

In mathematics , a Gromov hyperbolic space is a space with "evenly thin triangles". This term axiomatizes and generalizes spaces of negative curvature and has proven useful in many areas of mathematics.

definition

A geodesic triangle in a negatively curved surface

A geodesic metric space is called δ-hyperbolic for some δ≥0 if all geodesic triangles are δ-thin , i.e. H. every edge of the triangle is contained in the δ-neighborhood of the union of the other two edges:

{\ displaystyle [x, y] \ subseteq U _ {\ delta} ([y, z] \ cup [z, x]),}

{\ displaystyle [y, z] \ subseteq U _ {\ delta} ([z, x] \ cup [x, y]),}

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

This condition is, for example, for geodetic triangles in trees with or in the hyperbolic plane with met, generally for geodetic triangles in simply connected negative Riemannian manifolds sectional curvature . ${\ displaystyle \ delta = 0}$ ${\ displaystyle \ delta = ln ({\ sqrt {2}} + 1)}$

A δ-thin triangle

A metric space is called Gromov hyperbolic if it is δ-hyperbolic for some δ≥0.

Equivalently, hyperbolicity can be defined using the Gromov product . A metric space is δ- hyperbolic if it holds for all p , x , y and z in X.

{\ displaystyle (x, z) _ {p} \ geq \ min {\ big \ {} (x, y) _ {p}, (y, z) _ {p} {\ big \}} - \ delta .}

The δ hyperbolicity with regard to the first definition is equivalent to the δ hyperbolicity with regard to the second definition with a possibly different value of the constant δ .

Hyperbolic groups

A hyperbolic group is a finitely generated group whose Cayley graph for a finite generating system is δ-hyperbolic for a δ> 0. (Except for the constant δ, this condition is independent of the choice of the finite generating system.)

Gromov edge

The Gromov boundary of a δ- hyperbolic metric space is defined as the set of equivalence classes of sequences with respect to the equivalence relation ${\ displaystyle \ partial _ {\ infty} X}$ ${\ displaystyle X}$ ${\ displaystyle (x_ {n}) _ {n \ in N} \ subset X}$

{\ displaystyle (x_ {n}) _ {n \ in N} \ sim (y_ {n}) _ {n \ in N} \ Longleftrightarrow \ lim (x_ {n}, y_ {n}) _ {o} = \ infty}

for any (fixed) base point . ${\ displaystyle o \ in X}$

The topology of the Gromov boundary is determined by the environment base consisting of the sets

{\ displaystyle U (\ xi, r): = \ left \ {\ eta \ in \ partial _ {\ infty} X: \ exists (x_ {i}), (y_ {j}) \ sd \ \ xi = \ left [x_ {i} \ right], \ eta = \ left [y_ {j} \ right], \ liminf _ {i, j \ to \ infty} (x_ {i}, y_ {j}) _ { o} \ geq r \ right \}}

with . ${\ displaystyle \ xi \ in \ partial _ {\ infty} X, r \ geq 0}$

The Gromov product can be turned into a continuous function

{\ displaystyle (.,.) _ {o} \ colon \ partial _ {\ infty} X \ times \ partial _ {\ infty} X \ to \ left [0, \ infty \ right]}

continue.

literature

Sur les groupes hyperboliques d'après Mikhael Gromov. Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. Edited by É. Ghys and P. de la Harpe. Progress in Mathematics, 83. Birkhauser Boston, Inc., Boston, MA, 1990. ISBN 0-8176-3508-4