RESEARCH STATEMENT MENG-CHE \TURBO" HO The interplay between logic and geometric group theory has played a crucial role in both areas for many decades. My research program focuses on the interaction of these two rich topics, specifically on complexity notions with a computability-theoretic or combinatorial flavor in finitely-generated groups. I believe that the understanding of these aspects of finitely-generated groups will greatly help us in the grand process of categorizing classes of finitely-generated groups. Following this broad theme, my research breaks into four projects: Random groups. In an attempt to answer the question \What does a typical group look like?", Gromov defined the model for a random group, and showed that a random group is hyperbolic. This gives a nice formalization for understanding which properties of groups show up more often. Dunfield and Thurston also used similar ideas to give a model of random 3-manifolds, in an attempt to understand the virtual Haken conjecture. On the one hand, I am interested in different models of random groups. For instance, with a team of collaborators, we gave a model of random nilpotent group and studied its properties. On the other hand, I am also working on Knight's conjecture with Coulon and Logan, which states that the the first-order theory of random groups coincide with that of the free groups. In this work, We use the machinery developed by Sela in his solution to Tarski's problem extensively. Computable structure theory of groups. The goal of computable structure theory is to study the computability properties of mathematical structures. Since Maltsev and Novikov, group theory has been a testing ground for computability theory. One justification of this is the fact that groups are as hard as any class of mathematical structures. With Harrison-Trainor, I showed that finitely- generated groups are as hard as any class of finitely-generated structures. In addition to this, I also have result concerning the complexities of specific notions in group theory, including the descriptive complexity of a finitely-generated group, the hardness of cancelability of groups, and complexities of homomorphisms of groups. Word maps. Word maps, defined by substitution of formal words, are the analogue of polynomials in group theory. One focus is on word maps in finite simple groups, as seen in the Ore conjecture and Waring-type problems in groups. My work complement these results by focusing on word maps in nilpotent groups. In particular, with Cocke, I have studied images and fibers of word maps, and used our results to give an alternative characterization of finite nilpotent groups. Some of my work also concerns multilinear word maps in nilpotent and solvable groups, which has applications to group-based cryptographic protocols. Languages in groups. Formal languages were introduced by Chomsky as a tool to study human languages, and they are also useful in studying programming languages. It turns out the many natural sets in groups, when viewed as formal languages, have very nice properties, and are powerful tools in the study of groups. On the one hand, I am interested in characterizing groups with multiple context-free word problem, which will be the analogue of a theorem by Muller-Schupp. On the other hand, I also study geodesic representatives in metabelian groups, and use them to determine the rationality of the word growth and conjugacy growth of the groups. 1 1. Random groups There are different notions of randomness across mathematics, and they are useful in many ways. On the one hand, random constructions give us a precise way to talk about the generic behavior of a class of structures, and thus can be used as a test case for important conjectures; for instance Dunfield and Thurston [23] studied random 3-manifolds in order to understand the virtual Haken conjecture. On the other hand, random constructions are good at giving examples; for instance the Erd}os-R´enyi random graph [24] gives very good lower bounds on the Ramsey numbers. In the case of random groups, there are many competing models, and one of the most classical model given by Gromov is the density model with density d. For this model, Gromov showed the fundamental theorem of random groups, which says that a random group is infinite torsion-free hyperbolic if d < 1~2, and is trivial if d > 1~2. Another important result is Zuk's_ theorem [78, 52] that random groups in the density model with 1~3 < d < 1~2 have Kazhdan's property (T). Gromov [32] also used techniques from the theory of random groups to construct a finitely-presented group that admits no uniform embedding into the Hilbert space. One of my contribution in this area is to study random nilpotent groups [17] with a team of collaborators. This model is obtained by taking the density model at d = 0, then quotienting out a term in the lower central series of it. We focused on finding the distribution of ranks of random nilpotent groups, which generalizes the well-studied topic of random abelian groups by Dunfield-Thurston [23, §3.14], Kravchenko-Mazur-Petrenko [53], and Wang-Stanley [77], and has some interesting connections to number theory. Our model is different from theirs, but the same distribution of ranks is obtained. Theorem 1 ([17]). For a random m-generated, r-related nilpotent group G, the following probabil- ities are explicitly given: (1) The probability of G having rank less than m. (2) When r = m − 1 or m, the probability of G being cyclic. (3) When r ≤ m, the probability of G being abelian. (4) The probability of G being trivial. Many of our results can be lifted to gain information about the classical random group, for instance, we showed that for any d > 0, a random group is perfect in the density model. Our work was also generalized in [28], where they showed that in a random nilpotent group, the nilpotency class is as high as it can be, and the equation problem is as hard as it can be. In another work in progress with Coulon and Logan, I study the model theory of a random group, in particular, we work on Knight's conjecture: Conjecture 2. A first-order sentence is true in a free group if and only if it is true in a random group. Tarski's problem, resolved by Sela [73] and independently Kharlampovich and Myasnikov [50], states that the rank of the free group does not change the theory. One key idea in Sela's proof is the limit groups [72]. The idea of limit groups also sees application in acylindrically hyperbolic groups [34]. In order to answer Knight's conjecture, we study the limit groups over random groups, and use it to answer the 1-quantifier case in the positive: Theorem 3 ([18]). A first-order universal sentence is true in a free group if and only if it is true in a random group. 2. Computable structure theory of groups Historically, there have been many natural questions in group theory with a computability flavor, even outside the realm of logic. For example, Dehn [22] proposed the word problem for groups: For a given finitely-generated group, is there an algorithm that can determine if two words are equal 2 in the group? I am interested in the intrinsic computability strengths of groups as well as notions in group theory. The general form of the question can be formulated as: Given a notion in group theory, what is the complexity of the class of groups with this property? One basic notion in group theory is the isomorphism of groups, and its complexity can be measured by the complexities of Scott sentences of a group.1 In various work, the complexities of Scott sentences have been found for certain groups, including reduced abelian p-groups [47], free groups [8], finitely-generated abelian groups, the infinite dihedral group D∞, and certain torsion- free abelian groups of rank 1 [51]. Following this direction, I have determined the complexities of Scott sentences for polycyclic groups, solvable Baumslag-Solitar groups, lamplighter groups, wreath products of free abelian groups, and the free nilpotent group of countable rank [39]. Knight and Saraph [51] showed that every finitely-generated structure has a Σ3 Scott sentence. However, in all the examples mentioned above, this is not optimal and there is a d-Σ2 Scott sentence. Indeed, I also showed that cohopfian groups [39], which includes the random groups [31, 21, 71], 0 have the complexity for Scott sentences being d-Σ2. This result can be thought of as saying the Σ3 bound is not optimal in most groups. So, one wonders if there is any group where this bound is optimal. To answer this question, Harrison-Trainor and I found a model-theoretic characterization for the Σ3 Scott sentence to be optimal, and showed that such a group does exist. Theorem 4 ([36]). (1) Let A be a finitely-generated structure. Then A has a d-Σ2 Scott sen- tence if and only if there is no proper substructure B ⫋ A such that B ≅ A and B is Σ1-elementary in A. (2) There is a computable finitely-generated group that does not have a d-Σ2 Scott sentence. In particular, it has an optimal computable Σ3 Scott sentence. Independently, Alvir, Knight, and McCoy [2] showed that A has a d-Σ2 Scott sentence if and only if for some (or equivalently, all) generating tuples of A, the orbit of the tuple under Aut(A) is defined by a Π1 formula. As none of these conditions are group-theoretic in nature, one naturally asks if there is a group-theoretic characterization of a group with no d-Σ2 Scott sentence.