Advertisement call. A deep, repeated 5 note phrase that serves as a territorial song. This is the well know Woodpigeon song, often heard early in the morning.
Display flight. This is another territorial display most frequent in February and March. The male flies around the territory rising in a broad arc, and then gliding down with wings held stiffly horizontal advertising ownership. They can do the rising and falling a few times before alighting. Although mainly a visual advertising display, as the bird does not call, sometimes the bird will make a clapping noise with its wings at the apex of the arc.
The Bow display. It is a courtship display that territorial males perform when still unpaired and trying to attract a mate and also when the pair bond is becoming cemented. The male approaches the female and with the his neck is enlarged as he calls a 3 note call as he alternatively stretches its neck and then lowers its head, displaying the white and iridescent neck markings, and fanning the tail when it's highest. Check out this wonderful photo by Richard Hawley showing the display when fanning is at its maximum. Females can display some aggressiveness to the courting male, or move away or flee if they are not interested. This display often happens on trees, roofs or aerials, but occasionally on the ground. When on the ground the male will try and get closer and follow the female hopping to bow just in front of her.
Allopreening, billing and mating. Feeding the mate, gentle preening and billing precedes mating. I documented this behaviour in a series of photos here.
Slightly opened wings is a threat posture.aggressive stance, with neck stretched, flattened plumage and semi-open wings, and often this is enough to drive the intruder away. Occasionally, this develops into a fights, which sometimes occur in the depths of a tree. The pigeons will land near each other, alight their bodies and flap trying to hit each other (above).
Nest calling. This is a call by the male signalling to his partner a suitable nest site. I've covered this on other posts (here and here) and it is described in detail in the article by Cramp cited below. The call is harsh, and guttural composed of two notes, here is a recording of the call. I want to share here a video I took this morning, in which I got a good angle. You can see pecking or nodding movements to the nest floor in between calls:
Cramp, S. Territorial and other Behaviour of the Woodpigeon. Bird Study 5, 55–66 (1958). here.