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  "body": "\n## Statement\n\nA [[presentation|0JGEQXQ4AC]] $\\langle x_1, \\ldots, x_n \\mid r_1, \\ldots, r_n \\rangle$ is *balanced* if it has as many relators as generators.\nThe *Andrews–Curtis moves* on such a presentation, with $r_i$ elements of the [[free group|0KTW26C80J]] $F$ on $x_1, \\ldots, x_n$, are:\n\n1. replace some $r_i$ by $r_i^{-1}$;\n2. replace some $r_i$ by $r_i r_j$ for some $j \\ne i$;\n3. replace some $r_i$ by $g r_i g^{-1}$ for some $g \\in F$.\n\nThe conjecture: every balanced presentation of the trivial group can be turned into the trivial presentation $\\langle x_1, \\ldots, x_n \\mid x_1, \\ldots, x_n \\rangle$ by a finite sequence of Andrews–Curtis moves.\n\n## Remarks\n\nEach move keeps the normal closure of the relators, so it keeps the group.\nA presentation of the trivial group with $n$ generators has at least $n$ relators, since the abelianization $\\Z^n$ must become trivial; so the balanced ones are the extreme case.\nA weaker stable form of the conjecture also allows adding a generator $x_{n+1}$ with relator $x_{n+1}$, and removing such a pair.\nAndrews and Curtis stated the conjecture in 1965, in connection with the topology of $2$-dimensional complexes and $4$-dimensional manifolds.\nIt is open; many experts expect it to be false, and several families of balanced presentations of the trivial group are studied as potential counterexamples.\n\n## Explanations\n\n### L1 human\n\nWrite the trivial group with $n$ generators and $n$ rules in a complicated way.\nThe conjecture says that one can always simplify the rules to the obvious ones, $x_1 = 1, \\ldots, x_n = 1$.\nOnly three simple moves are allowed: invert a rule, multiply a rule by another, or conjugate a rule.\nFor example, $\\langle x, y \\mid x y^{-1}, y \\rangle$ becomes trivial: multiply the first rule by the second to get $x$.\nThe difficulty is that the needed sequence of moves may be extremely long or may not exist at all.\n",
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      "first_stated": "1965",
      "posed_by": "James J. Andrews and Morton L. Curtis"
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