# Groups with isomorphic factors OA-2-H0-to-H4

## Setting

You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.

## Problem

Prove 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 \(g\in F\).

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.