When it comes to finding synonyms for the phrase "giving someone the ax," there are a few different options that you can consider. One common alternative is to use the phrase "laying someone off," which is often used in business contexts to describe the process of terminating an employee's employment. Other options might include "terminating," "firing," or "dismissing" an employee. Depending on your context and the tone of your writing, you may want to choose a more formal or informal synonym for this phrase, so be sure to consider the connotations of each option before making your selection.