Geometric group action

In mathematics, specifically geometric group theory, a geometric group action is a certain type of action of a discrete group on a metric space.

Definition

In geometric group theory, a geometry is any proper, geodesic metric space. An action of a finitely-generated group G on a geometry X is geometric if it satisfies the following conditions:

Uniqueness

If a group G acts geometrically upon two geometries X and Y, then X and Y are quasi-isometric. Since any group acts geometrically on its own Cayley graph, any space on which G acts geometrically is quasi-isometric to the Cayley graph of G.

Examples

Cannon's conjecture states that any hyperbolic group with a 2-sphere at infinity acts geometrically on hyperbolic 3-space.

References

Geometry (disambiguation)

Geometry is a branch of mathematics dealing with spatial relationships.

Geometry or geometric may also refer to:

Geometry (Robert Rich album)

Geometry (1991) is an album by the American ambient and electronic musician Robert Rich. Although completed in 1988, this album was not released until three years later.

This album is more active and structured than any of his previous works. The music was inspired in part by the complex patterns of Islamic designs like those found at the Alhambra, and features complex just intonation. The first three tracks consist of complex layered sequences of electronic notes in rich, organic-sounding chime tones. Rich revisited this style in Gaudí (1991) and Electric Ladder (2006). Tracks 4 through 7 are slow textures more common to Robert Rich’s work. The album ends with another active piece similar to the first three tracks.

Tracks 1, 2, 4 and 8 were mixed by Robert Rich and future collaborator Steve Roach at Roach’s studio in Venice, California. This album was released in a two CD set with Numena in 1997.

Track listing

Apex

Apex may refer to:

Astronomy

Biology

Places

Organizations

Apex (headdress)

The apex was a cap worn by the flamines and Salii at Rome. The essential part of the apex, to which alone the name properly belonged, was a pointed piece of olive-wood, the base of which was surrounded with a lock of wool. This was worn on the top of the head, and was held there either by fillets only, or, as was more commonly the case, was also fastened by means of two strings or bands, which were called apicula (Festus, s.v.), or offendices (Festus, s.v.), though the latter word is also interpreted to mean a kind of button, by which the strings were fastened under the chin (cf. Serv. ad Virg. Aen. ii.683, viii.664, x.270).

The flamines were forbidden by law to go into public, or even into the open air without the apex (Gellius x.15), and hence we find the expression of alicui apicem dialem imponere used as equivalent to the appointment of a Flamen Dialis (Livy vi.41). Sulpicius was deprived of the priesthood, only because the apex fell from his head whilst he was sacrificing (Valerius Maximus i.1 §5).

Apex (mollusc)

In anatomy, an apex (adjectival form: apical) is part of the shell of a mollusk. The apex is the pointed tip (the oldest part) of the shell of a gastropod, scaphopod, or cephalopod.

Gastropods

The word "apex" is most often used to mean the tip of the spire of the shell of a gastropod. The apex is the first-formed, and therefore the oldest, part of the shell.

To be more precise, the apex would usually be where the tip of the embryonic shell or protoconch is situated, if that is still present in the adult shell (often it is lost or eroded away).

Coiled gastropod shells

The phrase apical whorls, or protoconch, means the whorls that constitute the embryonic shell at the apex of the shell, especially when this is clearly distinguishable from the later whorls of the shell, otherwise known as the teleoconch.

Comparison of the apical part and the whole shell of Otukaia kiheiziebisu:

Limpet-like gastropod shells

Where this feature is present, the space under the apex of a patellate or patelliform (limpet-like) gastropod shell is called the apical cavity.