Yes they do. Adidas' polo t-shirt may not be as iconic as the Ralph Lauren or Lacoste polo t-shirts, but it exists, and is fairly inexpensive: costing in the region of £16-21. As the images below show, the Adidas polo t-shirt is a simple design - available in white or black - that features the trefoil logo on the left chest. Polo t-shirts, while being a casual article of clothing, are generally viewed as being 'smart casual' and are deemed suitable for 'well heeled' sports like tennis and golf.
Adidas polo t-shirts are made from cotton and come with ribbed collar / sleeves and branded buttons. The t-shirt is also machine washable. There is little to distinguish the Adidas polo t-shirt from other polo t-shirts: except for the trefoil logo; which was released in 1971. This is not uncommon: polo t-shirts tend to feature plain cotton fabric; with a discreet crest/logo on the left chest (for branded apparel). History indicates that the polo t-shirt was invented when the British Military started to alter their polo shirts when stationed in the extreme summer heat of India.
John E Brooks, of the Brooks Brothers firm, was inspired to create the first US version of the polo shirt; after a trip to the UK in the 1900's. Not long after that the shirt became international: Jean Rene Lacoste making one of the first French versions of the shirt; incidentally, Dwight Eisenhower wore Lacoste polo shirts to play golf in the 1950's. By the 1970's, Fred Perry polo t-shirts were worn on football terraces and became an iconic piece of youth clothing. Present day, most fashion brands have their own version of the polo t-shirt, the most renown being the: Ralph Lauren, Lacoste and Fred Perry t-shirts (sorry Adidas).