{
  "statement": "H5: Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThe groups are not isomorphic as abstract groups:\n\\[\n\\Gamma_1\\not\\cong\\Gamma_2.\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 \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThe groups are not isomorphic as abstract groups:\n\\[\n\\Gamma_1\\not\\cong\\Gamma_2.\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-H5-to-H6",
  "given_context": {
    "premise": "Let \\(k=\\mathbb F_2\\), \\(R=k[t]\\), \\(A=R^3\\), and \\(C=(A\\otimes_k A)^{\\rm Flip}\\), where \\({\\rm Flip}(a\\otimes b)=b\\otimes a\\). All tensor products are over \\(k\\), and the vector spaces are regarded as discrete additive groups.\nOn the symmetric tensor basis associated with \\(\\mathcal B=\\{t^ne_j:n\\ge0,\\ 1\\le j\\le3\\}\\), define the \\(k\\)-linear map \\(\\delta:C\\to A\\) by\n\\[\n\\delta(b\\otimes b)=b,\\qquad\n\\delta(b\\otimes b'+b'\\otimes b)=0\\quad(b\\ne b').\n\\]\nLet \\(V=k^4\\), with coordinate basis \\(v_1,\\ldots,v_4\\), and set\n\\[\n\\omega(v,w)=\\sum_{j=1}^2(a_jb'_j+b_ja'_j),\\qquad\nr_0(v)=\\sum_{j=1}^2a_jb_j,\\qquad Q={\\rm Sp}(V,\\omega),\n\\]\nwhere \\(v=(a_1,b_1,a_2,b_2)\\), \\(w=(a'_1,b'_1,a'_2,b'_2)\\).\nFor \\(q\\in Q\\), define \\(\\ell_q\\in V^*\\) by\n\\[\n\\ell_q\\left(\\sum_{j=1}^4x_jv_j\\right)=\\sum_{j=1}^4x_jr_0(q^{-1}v_j).\n\\]\nPut \\(D=(A\\otimes_k V^*)\\oplus C\\), \\(H={\\rm SL}_3(R)\\times Q\\), and, for \\(i=1,2\\),\n\\[\n\\theta_i(l,q)(u,c)=\\bigl((l,q)u+(i-1)l\\delta(c)\\otimes\\ell_q,\\,lc\\bigr).\n\\]\nHere \\(lc=(l\\otimes l)c\\), \\((l,q)(a\\otimes\\varphi)=la\\otimes(q\\varphi)\\), and \\((q\\varphi)(v)=\\varphi(q^{-1}v)\\).\nDefine \\(\\Gamma_i\\) by generators \\(d\\in D\\), \\(h\\in H\\), the additive relations of \\(D\\), the multiplication relations of \\(H\\), and \\(hdh^{-1}=\\theta_i(h)d\\).\n\nThe groups are not isomorphic as abstract groups:\n\\[\n\\Gamma_1\\not\\cong\\Gamma_2.\n\\]"
  },
  "protocol": "single"
}