As far as I know, the resolution from the Gemini via its HDMI adapter is "(o) auto" only, which seems to be its native 2160x1080.
I have, thus far, tried the HDMI adapter with two TV sets. One of them is a regular 1920x1080 model. It says it's receiving 1080p @ 60Hz, and since it scales the image a bit, leaving small black bars top and bottom, I'd say it's 2160x1080p @ 60Hz. If I connect the Gemini to it via a Miracast receiver (an old Netgear P2TV 3000), it still says 2160x1080p @ 60Hz. As the actual HDMI signal, in this case, is generated by the Miracast receiver, it may depend of the model used, but I don't think I can change the resolution on mine anyway.
The other TV has a somewhat odd LCD panel (32" 1366x768 with its RGB stripes turned 90 degrees). It, too, says 1080p (but doesn't specify a frame rate) and, by default, scales the image to fit the width, with bars top and bottom. I haven't tried that one with the Miracast device, but I'd expect the same result.
One thing to note, which I've written about here:
www.oesf.org/index.php?act=findpost&pid=286176 is that both connection methods has a bit of lag and, unfortunately, doesn't delay audio and video equally. In my experiments, HDMI gives me 100-120ms video lag and 150ms audio lag, while Miracast gives me about 380-400ms video lag and 300ms audio lag. Depending on whether you need audio in sync with the video or not, that might be a bigger issue (it is for me, anyway) than the resolution, as most (though pehaps not all) modern display devices will likely be able to handle the 1080p signal.