In
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
...
theory, a thin group is a discrete
Zariski-dense subgroup of ''G''(R) that has infinite covolume, where ''G'' is a
semisimple algebraic group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direc ...
over the
reals. This is in contrast to a
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an ornam ...
, which is a discrete subgroup of finite covolume.
The theory of "group expansion" (
expander graph
In graph theory, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander constructions have spawned research in pure and applied mathematics, with several appli ...
properties of related
Cayley graph
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Ca ...
s) for particular thin groups has been applied to arithmetic properties of
Apollonian circles
In geometry, Apollonian circles are two families ( pencils) of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa. These circles form the basis for bipolar coordinates. T ...
and in
Zaremba's conjecture In mathematics, and more particularly in the analytic theory of regular continued fractions, an infinite regular continued fraction ''x'' is said to be ''restricted'', or composed of restricted partial quotients, if the sequence of denominators of ...
.
References
*
Linear algebraic groups
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