In my opinion, the term "hybrid" is both loosely used and vague. It can mean so many different things. But in terms of a Giant Escape 3, I would think it would quite a bit quicker than your current bike and assuming your level of fitness is similar to the people you ride with, you should be able to easily keep up with people on fixies, if not run circles around them.

If you look at my two bikes, my older Jamis has front suspension, a spring in the seat post, 100psi 700 x 32mm tires and 3 x 8 gears. My newer Giant no suspension, 120psi 700 x 28mm tires and 2 x 10 gearing. Both of these bikes are considered hybrids. The Jamis being more of a "city" bike and the Giant more a "sports fitness" bike. Even though the size and geometry of the two bikes is very similar, I can assure you they are very different bikes and the Giant is the much "faster" of the two bikes. The first couple times I rode my new Giant, it shocked me how easily it gained speed when going down any sort of hill.