\(\mathrm{SL}(2,\mathbb Q_p), p\neq 2\), has no nontrivial continuous finite-dimensional pseudorepresentations (Q2918426)

For a given topological group \(G\), a \textit{quasirepresentation} (more precisely, an \(\varepsilon\)-quasirepresentation) is a map from \(G\) into the group of invertible operators in a Banach space \(E\) such that NEWLINE\[NEWLINE\|\pi(g_1g_2)-\pi(g_1)\pi(g_2)\| \leq \varepsilon, \forall g_1,g_2 \in G,NEWLINE\]NEWLINE for some small \(\varepsilon > 0\). A quasirepresentation is called a \textit{pseudorepresentation} if the image of any positive integer power of the element of the group is similar to the corresponding positive integer power of the element by means of an operator belonging to a small operator-norm neighborhood of the identity operator.NEWLINENEWLINEThe main result of the paper is Theorem 2: every continuous finite-dimensional pseudorepresentation of \(\mathrm{SL}(2,\mathbb Q_p), p\neq 2\), is trivial, i.e. is a multiple of the one-dimensional identity representation of \(\mathrm{SL}(2,\mathbb Q_p)\). This supports the conjecture that the simple Chevalley groups over totally disconnected locally compact fields of characteristic zero have no nontrivial continuous finite-dimensional pseudorepresentations.