Final answer:
Interest groups influence the Supreme Court mainly by filing amicus curiae briefs, presenting arguments about how a decision may impact groups or the general public.
Step-by-step explanation:
One of the main ways that interest groups attempt to influence the Supreme Court's decisions is by filing amicus curiae briefs. These 'friend of the court' briefs are legal documents that present arguments and the potential impacts of the Court's decisions on the population or on specific groups.
This method allows interest groups to educate and persuade the justices on issues, even though the groups are not direct parties in the case. Notable examples include the file of amicus briefs by the Human Rights Campaign in Obergefell v. Hodges, a landmark decision where the Supreme Court legalized same-sex marriage nationwide.