Welcome to you, Allanwski, too!
Another tidbit of info, the place is ripe with recent history, having been theater of WWI fighting between Italian and Austro-Hungarian troops
(there were some Allied troops on, mainly Birtish and American, on the Italian side and Germans on the Austrian one).
Lagazuoi peak was blown apart by an Italian and Austrian mines in 1915-1916.
Marmolada is also a must to ski, and there too one can visit a WWI museum...The Lagazuoi one is best visited in summer, since it's "open air"...
If you happen to ski on "Col di Lana" (dubbed the "bloody hill") you'll be skiing on one of the worst contended places of the area
Skiing there, to Italians and Austrians (even if very few think about it, I try never forget to say at least a prayer)...is like going to visit WWI Belgian battlefields or Normandy beaches for others...
Out of the Dolomites, the Passo Tonale, to the east is also such a place. In all these area one can see tranches, tracks and shelters dug in the rocks with only shovels and pickaxes...I'll borrow from Simonides
"O viandante, annuncia agli Spartani che qui
noi morimmo obbedienti al loro comando"
"Go tell the Spartans, thou who passest by,
That here, obedient to their laws, we lie"