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If you could post a picture of all the permissions that global moderators on your board have, I could tell you a better answer. But even if you don't give them the permission of direct administration, certain permissions are still located in the admin panel.

   A screenshot of what HE sees in Admin would be helpful.

The best way for you to find out exactly what abilitys he does have is to create a dummy user for yourself to use.  Set that user as a Global Moderator then login as that user and go around an check to see that the user has only the options/abilitys you want the user to have.  That way you can also go into Admin yourself (as the dummy user) and see it as they see it.

Change the view in Permissions to Classic and look thru all settings for Global Moderator.  If you have only Access the moderation center checked for him then that is all he should be able to access via the tab on the menu bar.    Are you sure you do not have him set as Administrator as a Secondary Membergroup?   Go to his profile, Account Settings - click on "Show Additonal Groups" and see if you have him set as Admin there.

Ive set up a test user on my test forum and set the user as Global Mod using the default permissions for that group.   There is NO Admin tab showing for my test user so either you have set him as Admin as a secondary membergroup or have checked something in Permissions for that group that is allowing him to see the Admin tab.