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I finally figured it out.  Jorge's answer was almost correct, but I had to manually create an Environment variable for ExecutionImpl.getActivity() to succeed.  So the final code was

EnvironmentFactory environmentFactory = (EnvironmentFactory)processEngine;
    Execution execution = executionService.findExecutionById(executionID);
    if (execution instanceof ExecutionImpl) {
     ExecutionImpl executionImpl = (ExecutionImpl)execution;
     EnvironmentImpl environment = environmentFactory.openEnvironment();
     ActivityImpl activityImpl = null;
     try {
      activityImpl = executionImpl.getActivity();
      } finally {
      environment.close();
     } 

     if (activityImpl != null) {   
         return activityImpl.getOutgoingTransitionsMap().keySet();

     }
    }    

    return Collections.<String>emptySet();