{
  "statement": "H3: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(V=k^4\\). All tensor products are over \\(k\\). Put\n\\[\nC=(A\\otimes_k A)^{\\rm Flip},\\quad \\Delta(a)=a\\otimes a,\\quad\nD=(A\\otimes_k V^*)\\oplus C,\n\\]\nwhere \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\), and regard the vector spaces as discrete additive groups.\nThe group \\({\\rm SL}_3(R)\\) acts on \\(A\\) by its usual action, on \\(C\\) by the tensor action, and trivially on \\(V^*\\).\n\nLet \\(\\widehat D={\\rm Hom}_k(D,k)\\) with the compact product topology, identified with the character group by \\(\\chi\\mapsto(d\\mapsto(-1)^{\\chi(d)})\\). The \\({\\rm SL}_3(R)\\)-action is \\(l\\chi=\\chi\\circ l^{-1}\\), where \\(l(u,c)=(lu,lc)\\).\nFor any basis \\(\\varphi_1,\\ldots,\\varphi_4\\) of \\(V^*\\), set\n\\[\nd_j=(e_1\\otimes\\varphi_j,0)\\quad(1\\le j\\le4),\\qquad\nd_5=(0,\\Delta(e_1)).\n\\]\nFor every \\({\\rm SL}_3(R)\\)-invariant Borel probability measure \\(\\mu\\) on \\(\\widehat D\\),\n\\[\n\\mu(\\widehat D\\setminus\\{0\\})\\le12\\sum_{j=1}^5\\mu\\{\\chi:\\chi(d_j)=1\\}.\n\\]\n\nProve H6: A discrete group has property \\((T)\\) if every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), there is a unit vector \\(\\xi\\) with \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).\nA group is ICC if the conjugacy class of every nonidentity element is infinite.\n\nThere exist two countable ICC groups \\(\\Gamma\\) and \\(\\Lambda\\), both with property \\((T)\\), which are not isomorphic as abstract groups, but whose group von Neumann algebras are trace-preservingly normally \\(*\\)-isomorphic \\({\\rm II}_1\\) factors:\n\\[\n\\Gamma\\not\\cong\\Lambda,\\qquad L(\\Gamma)\\cong L(\\Lambda).\n\\]\nThe algebra \\(L(\\Gamma)\\) is generated by the left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\); \\(L(\\Lambda)\\) is defined in the same way. A \\({\\rm II}_1\\) factor is an infinite-dimensional von Neumann algebra with center \\(\\mathbb C I\\) and a faithful normal tracial state.",
  "starting_statement": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(V=k^4\\). All tensor products are over \\(k\\). Put\n\\[\nC=(A\\otimes_k A)^{\\rm Flip},\\quad \\Delta(a)=a\\otimes a,\\quad\nD=(A\\otimes_k V^*)\\oplus C,\n\\]\nwhere \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\), and regard the vector spaces as discrete additive groups.\nThe group \\({\\rm SL}_3(R)\\) acts on \\(A\\) by its usual action, on \\(C\\) by the tensor action, and trivially on \\(V^*\\).\n\nLet \\(\\widehat D={\\rm Hom}_k(D,k)\\) with the compact product topology, identified with the character group by \\(\\chi\\mapsto(d\\mapsto(-1)^{\\chi(d)})\\). The \\({\\rm SL}_3(R)\\)-action is \\(l\\chi=\\chi\\circ l^{-1}\\), where \\(l(u,c)=(lu,lc)\\).\nFor any basis \\(\\varphi_1,\\ldots,\\varphi_4\\) of \\(V^*\\), set\n\\[\nd_j=(e_1\\otimes\\varphi_j,0)\\quad(1\\le j\\le4),\\qquad\nd_5=(0,\\Delta(e_1)).\n\\]\nFor every \\({\\rm SL}_3(R)\\)-invariant Borel probability measure \\(\\mu\\) on \\(\\widehat D\\),\n\\[\n\\mu(\\widehat D\\setminus\\{0\\})\\le12\\sum_{j=1}^5\\mu\\{\\chi:\\chi(d_j)=1\\}.\n\\]",
  "target_statement": "A discrete group has property \\((T)\\) if every unitary representation with almost invariant unit vectors has a nonzero invariant vector. Almost invariant means that for every finite set \\(F\\) of group elements and every \\(\\varepsilon>0\\), there is a unit vector \\(\\xi\\) with \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).\nA group is ICC if the conjugacy class of every nonidentity element is infinite.\n\nThere exist two countable ICC groups \\(\\Gamma\\) and \\(\\Lambda\\), both with property \\((T)\\), which are not isomorphic as abstract groups, but whose group von Neumann algebras are trace-preservingly normally \\(*\\)-isomorphic \\({\\rm II}_1\\) factors:\n\\[\n\\Gamma\\not\\cong\\Lambda,\\qquad L(\\Gamma)\\cong L(\\Lambda).\n\\]\nThe algebra \\(L(\\Gamma)\\) is generated by the left regular representation on \\(\\ell^2(\\Gamma)\\), with canonical trace \\(\\tau(\\lambda_g)=\\mathbf1_{g=1}\\); \\(L(\\Lambda)\\) is defined in the same way. A \\({\\rm II}_1\\) factor is an infinite-dimensional von Neumann algebra with center \\(\\mathbb C I\\) and a faithful normal tracial state.",
  "case_id": "OA-2-H3-to-H6",
  "given_context": {
    "premise": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(V=k^4\\). All tensor products are over \\(k\\). Put\n\\[\nC=(A\\otimes_k A)^{\\rm Flip},\\quad \\Delta(a)=a\\otimes a,\\quad\nD=(A\\otimes_k V^*)\\oplus C,\n\\]\nwhere \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\), and regard the vector spaces as discrete additive groups.\nThe group \\({\\rm SL}_3(R)\\) acts on \\(A\\) by its usual action, on \\(C\\) by the tensor action, and trivially on \\(V^*\\).\n\nLet \\(\\widehat D={\\rm Hom}_k(D,k)\\) with the compact product topology, identified with the character group by \\(\\chi\\mapsto(d\\mapsto(-1)^{\\chi(d)})\\). The \\({\\rm SL}_3(R)\\)-action is \\(l\\chi=\\chi\\circ l^{-1}\\), where \\(l(u,c)=(lu,lc)\\).\nFor any basis \\(\\varphi_1,\\ldots,\\varphi_4\\) of \\(V^*\\), set\n\\[\nd_j=(e_1\\otimes\\varphi_j,0)\\quad(1\\le j\\le4),\\qquad\nd_5=(0,\\Delta(e_1)).\n\\]\nFor every \\({\\rm SL}_3(R)\\)-invariant Borel probability measure \\(\\mu\\) on \\(\\widehat D\\),\n\\[\n\\mu(\\widehat D\\setminus\\{0\\})\\le12\\sum_{j=1}^5\\mu\\{\\chi:\\chi(d_j)=1\\}.\n\\]"
  },
  "protocol": "single"
}