Step 1: Warming a bromo compound with aqueous AgNO3 tests SN1 solvolysis rate; faster carbocation formation gives a faster precipitate.
Step 2: Compound I, bromobenzene, has a highly unstable non-delocalized phenyl cation on ionization; essentially unreactive.
Step 3: Compound II, allylic bromide, gives an allylic cation with modest resonance stabilization.
Step 4: Compound III ionizes to give the tropylium cation, a fully aromatic 7-membered ring cation with 6 delocalized pi electrons, exceptionally stable.
Step 5: Compound IV, 4-bromopyridine, has an even more destabilized aryl-type cation due to the electron-poor ring.
Compound III solvolyzes fastest.
\[\boxed{\text{III}}\]