Definitions of sway:

• noun:   controlling influence
• noun:   pitching dangerously to one side
• verb:   move back and forth or sideways

  Example: "The tall building swayed"

• verb:   cause to move back and forth

  Example: "The wind swayed the trees gently"

• verb:   move or walk in a swinging or swaying manner
• verb:   win approval or support for

  Example: "His speech did not sway the voters"