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  "body": "\n## Example\n\nThe group $SL(2, \\R)$ is the [[special linear group|0V9PMN0RBH]] of real $2 \\times 2$ matrices with determinant $1$:\n$$SL(2, \\R) = \\set{\\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix} : a, b, c, d \\in \\R, \\ ad - bc = 1}.$$\nIt is a closed subgroup of $GL_2(\\R)$ cut out by one equation, so it is a [[Lie group|7JBRGCVJRB]] of dimension $3$.\n\nIt acts on the [[upper half-plane|X6XGGH5FPW]] $\\mathbb{H} = \\set{z \\in \\C : \\operatorname{Im} z \u003e 0}$ by [[Möbius transformations|0SG74WBFA2]]\n$$z \\mapsto \\frac{az + b}{cz + d}.$$\nCheck that $\\mathbb{H}$ is preserved: multiplying numerator and denominator by $c\\bar{z} + d$ gives\n$$\\operatorname{Im} \\frac{az + b}{cz + d} = \\frac{(ad - bc) \\operatorname{Im} z}{\\abs{cz + d}^2} = \\frac{\\operatorname{Im} z}{\\abs{cz + d}^2} \u003e 0.$$\n\n## Remarks\n\nThe matrices $I$ and $-I$ act as the identity map, and the quotient $PSL(2, \\R) = SL(2, \\R)/\\set{\\pm I}$ is the group of orientation-preserving isometries of the hyperbolic plane in the upper half-plane model.\n$SL(2, \\R)$ is connected but not compact.\nIts maximal compact subgroup is the rotation group [[SO(2)|0CGYNDCWRR]], and $SL(2, \\R)$ is homeomorphic to $SO(2) \\times \\R^2$.\nSo its fundamental group is $\\Z$, and it is not simply connected.\nIts universal cover is a Lie group with no faithful finite-dimensional representation.\nAn element $g \\ne \\pm I$ is called elliptic, parabolic or hyperbolic when $\\abs{\\tr g}$ is less than, equal to, or greater than $2$.\nIts Lie algebra $\\mathfrak{sl}(2, \\R)$ of trace-zero real $2 \\times 2$ matrices is simple, so $SL(2, \\R)$ is a simple Lie group.\n\n## Explanations\n\n### L1 human\n\n$SL(2, \\R)$ is the group of linear maps of the plane that keep area and orientation.\nRotations, shears such as $\\begin{pmatrix} 1 \u0026 t \\\\ 0 \u0026 1 \\end{pmatrix}$, and squeezes such as $\\begin{pmatrix} e^t \u0026 0 \\\\ 0 \u0026 e^{-t} \\end{pmatrix}$ all belong to it.\nThe same group also moves the points of the upper half-plane by fractional linear maps, and these are the symmetries of the hyperbolic plane.\nIt is among the smallest noncompact simple Lie groups, and its study is the entry point to the representation theory of such groups.\n",
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