SL2(R) - Classification of Elements

Classification of Elements

The eigenvalues of an element A ∈ SL(2,R) satisfy the characteristic polynomial

and therefore

This leads to the following classification of elements, with corresponding action on the Euclidean plane:

• If | tr(A) | < 2, then A is called elliptic, and is conjugate to a rotation.
• If | tr(A) | = 2, then A is called parabolic, and is a shear mapping.
• If | tr(A) | > 2, then A is called hyperbolic, and is a squeeze mapping.

The names correspond to the classification of conic sections by eccentricity: if one defines eccentricity as half the absolute value of the trace (ε = ½ tr; dividing by 2 corrects for the effect of dimension, while absolute value corresponds to ignoring an overall factor of ±1 such as when working in PSL(2, R)), then this yields:, elliptic;, parabolic;, hyperbolic.

The identity element 1 and negative identity element -1 (in PSL(2,R) they are the same), have trace ±2, and hence by this classification are parabolic elements, though they are often considered separately.

The same classification is used for SL(2,C) and PSL(2,C) (Möbius transformations) and PSL(2,R) (real Möbius transformations), with the addition of "loxodromic" transformations corresponding to complex traces; analogous classifications are used elsewhere.

A subgroup that is contained with the elliptic (respectively, parabolic, hyperbolic) elements, plus the identity and negative identity, is called an elliptic subgroup (respectively, parabolic subgroup, hyperbolic subgroup).

This is a classification into subsets, not subgroups: these sets are not closed under multiplication (the product of two parabolic elements need not be parabolic, and so forth). However, all elements are conjugate into one of 3 standard one-parameter subgroups (possibly times ±1), as detailed below.

Topologically, as trace is a continuous map, the elliptic elements (excluding ±1) are an open set, as are the hyperbolic elements (excluding ±1), while the parabolic elements (including ±1) are a closed set.