Do abbreviated names need a period after them if they have a comma after?

I learned that when abbreviating a word, such as "tomorrow" as "tom.," we should place a period right after it. But what about when it's a name? For example, which sentence would be correct? (The abbreviated name is "Gavin.")

With the period:

"Gav., what are you doing this Saturday?"

Without the period:

"Gav, what are you doing this Saturday?"

Do abbreviate names need a period after them?

Solution 1:

Abbreviated names often do have periods in them, as in "Jos. A. Bank." This case is different from the one in the OP's example, which as some commenters have observed is probably a nickname rather than an abbreviation.

The Wiktionary list of Abbreviations for English Names claims that a period is left off if the abbreviation ends with the last letter of the name (as in "Thos" for Thomas). But surely this nuance depends on what style guide you are using.