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  "body": "\n## Description\n\nThompson's group $F$ is the [[group|G5S1M9]], under composition, of all [[homeomorphisms|0QT00PB5G2]] $f \\colon [0, 1] \\to [0, 1]$ such that:\n\n1. $f$ is piecewise linear, with finitely many break points;\n2. every break point is a dyadic rational, a number $k / 2^j$ with integers $k$ and $j \\ge 0$;\n3. every slope of $f$ is an integer power of $2$.\n\nThompson's group $T$ is the group of such piecewise linear maps of the circle $[0, 1] / \\set{0, 1}$, which also map the dyadic rationals onto themselves.\nThompson's group $V$ is the group of right-continuous bijections of $[0, 1)$ that are piecewise linear in the same sense.\nThey may have finitely many jumps, all at dyadic rationals.\nThe inclusions $F \\subset T \\subset V$ hold.\n\nFor example, the map $A$ with $A(x) = x / 2$ on $[0, 1/2]$, $A(x) = x - 1/4$ on $[1/2, 3/4]$ and $A(x) = 2x - 1$ on $[3/4, 1]$ lies in $F$.\nThe pieces agree at the break points: $1/2 \\mapsto 1/4$ and $3/4 \\mapsto 1/2$, and the slopes are $1/2$, $1$ and $2$.\n\n## Remarks\n\nAll three groups are [[finitely presented|TJKANTKASZ]].\n$F$ has the infinite presentation $\\langle x_0, x_1, x_2, \\ldots \\mid x_i^{-1} x_n x_i = x_{n+1} \\text{ for } i \u003c n \\rangle$, and it is generated by $x_0$ and $x_1$ with two relations.\nThe commutator subgroup $[F, F]$ is simple, and $F / [F, F] \\cong \\Z^2$.\n$F$ is torsion-free and contains no [[free subgroup|0KTW26C80J]] of rank $2$.\nWhether $F$ is [[amenable|CEAT9BESKK]] is a well-known open problem.\n$T$ and $V$ are infinite [[simple groups|0YMYMNQ0QZ]] that are finitely presented; they were the first known examples of such groups.\n\n## Explanations\n\n### L1 human\n\nCut the interval $[0, 1]$ into pieces by repeatedly halving, and do the same with another copy of $[0, 1]$, with the same number of pieces.\nStretch each piece of the first copy linearly onto the matching piece of the second.\nThe maps obtained this way form Thompson's group $F$.\nThe groups $T$ and $V$ allow the pieces to be rotated cyclically or shuffled in any order.\nThese simple-looking groups answer many questions in group theory and serve as counterexamples to many guesses.\n",
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