We study two complexity notions of groups -the syntactic complexity of a computable Scott sentence and the m-degree of the index set of a group. Finding the exact complexity of one of them usually involves finding the complexity of the other, but this is not always the case. Knight et al. determined the complexity of index sets of various structures.In this paper, we focus on finding the complexity of computable Scott sentences and index sets of various groups. We give computable Scott sentences for various different groups, including nilpotent groups, polycyclic groups, certain solvable groups, and certain subgroups of Q. In some of these cases, we also show that the sentences we give are optimal. In the last section, we also show that d-Σ 2 Δ 3 in the complexity hierarchy of pseudo-Scott sentences, contrasting the result saying d-Σ 2 = Δ 3 in the complexity hierarchy of Scott sentences, which is related to the boldface Borel hierarchy.

Scott showed that for every countable structure A, there is a sentence of the infinitary logic Lω 1 ω , called a Scott sentence for A, whose models are exactly the isomorphic copies of A. Thus, the least quantifier complexity of a Scott sentence of a structure is an invariant that measures the complexity "describing" the structure. Knight et al. have studied the Scott sentences of many structures. In particular, Knight and Saraph showed that a finitely generated structure always has a Σ 0 3 Scott sentence. We give a characterization of the finitely generated structures for whom the Σ 0 3 Scott sentence is optimal. One application of this result is to give a construction of a finitely generated group where the Σ 0 3 Scott sentence is optimal.We use this hierarchy to measure the complexity of a sentence. The sentence given above describing the infinite-dimensional Q-vector space is a Π 0 3 sentence, and it turns out that this is the best possible; there is no d-Σ 0 2 description of this vector space. There is a d-Σ 0 2 description of any finite-dimensional Q-vector space, and so these structures are "simpler" than the infinite-dimensional vectors space.

The word problem of a group {G=\langle\Sigma\rangle} can be defined as the set of formal words in {\Sigma^{*}} that represent the identity in G. When viewed as formal languages, this gives a strong connection between classes of groups and classes of formal languages. For example, Anīsīmov showed that a group is finite if and only if its word problem is a regular language, and Muller and Schupp showed that a group is virtually-free if and only if its word problem is a context-free language. Recently, Salvati showed that the word problem of {\mathbb{Z}^{2}} is a multiple context-free language, giving the first example of a natural word problem that is multiple context-free, but not context-free. We generalize Salvati’s result to show that the word problem of {\mathbb{Z}^{n}} is a multiple context-free language for any n.

Abstract. Word maps in a group, an analogue of polynomials in groups, are defined by substitution of formal words. In [Lub14], Lubotzky gave a characterization of the images of word maps in finite simple groups, and a consequence of his characterization is the existence of a group G such that the image of some word map on G is not closed under inversion. We explore sufficient conditions on a group that ensure that the image of all word maps on G are closed under inversion. We then show that there are only two groups with order less than 108 with the property that there is a word map with image not closed under inversion. We also study this behavior in nilpotent groups.