This plant is called the Mass Cane or Dracaena Massangea also known as Dracaena fragrans . See here for other answers which expand on this.
This plant appears to have been grown from a cutting which is why the stem is only an inch thick. When grown in tropical climates it can have a stem which is three to four inches (six to nine cm) thick.
It is rare for this plant to bud out and grow wider without some help. You can do this and get two stems from one:
- Cut the stem cleanly between two leaves.
- from the new cutting take some leaves from the bottom
- put the cutting in water in bright indirect sunlight
When the new cutting has roots that are at least a few inches long put it back in the pot or pot it up separately for a while and then pot them up together.