{
  "statement": "H4: 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\nBoth \\(\\Gamma_1\\) and \\(\\Gamma_2\\) have property \\((T)\\): 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\\), some unit vector \\(\\xi\\) satisfies \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).\n\nProve 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\\]",
  "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\nBoth \\(\\Gamma_1\\) and \\(\\Gamma_2\\) have property \\((T)\\): 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\\), some unit vector \\(\\xi\\) satisfies \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\).",
  "target_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\\]",
  "case_id": "OA-2-H4-to-H5",
  "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\nBoth \\(\\Gamma_1\\) and \\(\\Gamma_2\\) have property \\((T)\\): 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\\), some unit vector \\(\\xi\\) satisfies \\(\\|\\pi(g)\\xi-\\xi\\|<\\varepsilon\\) for all \\(g\\in F\\)."
  },
  "protocol": "single"
}