The Identity Problem in $\mathbb{Z}\wr \mathbb{Z}$ is decidable

We consider semigroup algorithmic problems in the wreath product $\mathbb{Z}\wr \mathbb{Z}$. Our paper focuses on two decision problems introduced by Choffrut and Karhumäki (2005): the Identity Problem (does a semigroup contain the neutral element?) and the Group Problem (is a semigroup a group?) for finitely generated sub-semigroups of $\mathbb{Z}\wr \mathbb{Z}$. We show that both problems are decidable. Our result complements the undecidability of the Semigroup Membership Problem (does a semigroup contain a given element?) in $\mathbb{Z}\wr \mathbb{Z}$ shown by Lohrey, Steinberg and Zetzsche (ICALP 2013), and contributes an important step towards solving semigroup algorithmic problems in general metabelian groups.