$G_1$ and $G_2$ are minimal lifts of $\bar G_1$ and $\bar G_2$. Note that $\delta(\rho_1, \rho_2)$ is unchanged if we factor through the common kernel $N_1 \cap N_2$, so we may assume $N_1 \cap N_2 = 1$. Then $N_1 \times N_2$ is a normal subgroup of $G$, $N_1 \simeq \rho_2(N_1) \lhd G_2$ and $N_2 \simeq \rho_1(N_2) \lhd G_1$.

Write $G_i = H_i \times_{Z(H_i)} Z_i$ for $i=1, 2$, where $H_i$ is a minimal lift of $\bar G_i$ to $\operatorname{GL}_n({\mathbb C})$ and $Z_i$ is a cyclic group containing $Z(H_i)$.

For a subgroup $H$ of $\operatorname{GL}_n({\mathbb C})$, let $\alpha_n(H)$ be the minimum of $\frac{ | \{ h \in H : \operatorname{tr}h \ne 0 \} | }{|H|}$ as one ranges over all embeddings (i.e., faithful $n$-dimensional representations) of $H$ in $\operatorname{GL}_n({\mathbb C})$.

\[lem:34\] Let $m = |\rho_1(N_2) \cap Z(G_1)|$. Then $\delta(\rho_1, \rho_2) \ge \frac{m-1}{m} \alpha_n(H_1)$.

Let $K = N_2 \cap \rho_1^{-1}(Z(G_1))$, so $\rho_1(K)$ is a cyclic subgroup of $Z(G_1)$ of order $m$ and $\rho_2(K) = 1$. Fix any $g \in G$. Then as $k$ ranges over $K$, $\operatorname{tr}\rho_1(gk)$ ranges over the values $\zeta \operatorname{tr}\rho(g)$, where $\zeta$ runs through all $m$-th roots of 1 in ${\mathbb C}$, attaining each value equally often. On the other 