# Groups with isomorphic factors OA-2-H4-to-H6

## Setting

You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.

## Problem

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.
On 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
\[
\delta(b\otimes b)=b,\qquad
\delta(b\otimes b'+b'\otimes b)=0\quad(b\ne b').
\]
Let \(V=k^4\), with coordinate basis \(v_1,\ldots,v_4\), and set
\[
\omega(v,w)=\sum_{j=1}^2(a_jb'_j+b_ja'_j),\qquad
r_0(v)=\sum_{j=1}^2a_jb_j,\qquad Q={\rm Sp}(V,\omega),
\]
where \(v=(a_1,b_1,a_2,b_2)\), \(w=(a'_1,b'_1,a'_2,b'_2)\).
For \(q\in Q\), define \(\ell_q\in V^*\) by
\[
\ell_q\left(\sum_{j=1}^4x_jv_j\right)=\sum_{j=1}^4x_jr_0(q^{-1}v_j).
\]
Put \(D=(A\otimes_k V^*)\oplus C\), \(H={\rm SL}_3(R)\times Q\), and, for \(i=1,2\),
\[
\theta_i(l,q)(u,c)=\bigl((l,q)u+(i-1)l\delta(c)\otimes\ell_q,\,lc\bigr).
\]
Here \(lc=(l\otimes l)c\), \((l,q)(a\otimes\varphi)=la\otimes(q\varphi)\), and \((q\varphi)(v)=\varphi(q^{-1}v)\).
Define \(\Gamma_i\) by generators \(d\in D\), \(h\in H\), the additive relations of \(D\), the multiplication relations of \(H\), and \(hdh^{-1}=\theta_i(h)d\).

Both \(\Gamma_1\) 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 \(g\in F\).

Prove 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 \(g\in F\).
A group is ICC if the conjugacy class of every nonidentity element is infinite.

There 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:
\[
\Gamma\not\cong\Lambda,\qquad L(\Gamma)\cong L(\Lambda).
\]
The 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.

Write the full proof to answer.md in the workspace root. Write the statements and hypotheses of any external results you cite to cited_results.md; if none are used, write 'No external results cited.' Identify any gap you cannot close.