Consider the group $G = \ast_{i=1}^k \mathbb Z_2 = \langle (g_i)_{i=1}^k : g_i^2 = 1 \rangle$. Suppose we define a homomorphism $\phi$ from $G$ to $O(3)$ by setting $\phi(g_i)$ to the reflection in a randomly chosen plane $H_i = \{x\in\mathbb R^3 : x\cdot \hat v_i = 0\}$, where the unit vectors $\hat v_i$ are iid and uniformly distributed over the sphere.
Can we show that $\phi$ is injective almost surely?
Someone else has written some code (for a programming competition) that relies on that assumption, but they did not describe a proof, and while the assumption seems intuitively obvious to me, I have been unable to produce a formal proof.
I'm also curious whether this is true in $O(2)$ or in higher dimensions.
[edit: It's false for $O(2)$ when $k > 2$. The identities at https://en.wikipedia.org/wiki/Rotations_and_reflections_in_two_dimensions imply any such $\phi$ will have $(g_1g_2g_3)^2$ in the kernel regardless of $\hat v_1,\hat v_2,\hat v_3$.]