TITLE: A limit of conjugate representations in $\mathrm{GL}(n,\mathbb C)$
QUESTION [5 upvotes]: $\DeclareMathOperator{\GL}{GL}$Let $G$ be a finitely presented group with generators $g_1,\dotsc, g_k$. Suppose we have a family of representations $\rho_t:G\to \GL(n,\mathbb C)$ with $t\in [0,1]$ smoothly dependent on $t$. Suppose that  for any $t\in [0,1]$ and $i\in\{1,\dotsc,k\}$, $\rho_t(g_i)$ is diagonalisable, and for $t>0$ all representations $\rho_t$ are conjugate.
Question. Is it true that $\rho_0$ is conjugate to $\rho_t$ for $t>0$?
(PS Yves gave a counter-example in a comment.)
Is the answer positive, if we additionally ask $\rho_t(G)\subset \GL(n,\mathbb C)$ to be conjugate to a subgroup of $\operatorname U(n)\subset \GL(n,\mathbb C)$ for all $t$?

REPLY [7 votes]: It is proven in Lemma 1.25 of the book "Varieties of representations" by Lubotzky and Magid (Memoirs of AMS, vol. 336, 1985) that each $GL(n,{\mathbb C})$-conjugation orbit of a semisimple representation $\rho\in R_n={\mathrm Hom}(\Gamma, GL(n, {\mathbb C}))$ is Zariski-closed in $R_n$. If a representation is unitarizable, then it is completely reducible, i.e. semisimple. Thus, you get the property you are after. Note that you do not need finite presentability of $\Gamma$, only finite generation.