Research Statement

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-Rényi 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 sentence 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. Our construction involves coding a graph with certain unary operations into a finitely-generated group via small cancellation theory and HNN extensions. However, this inevitably produces groups that are not finitely-presented. As a result, the following conjecture remains open: 0 Conjecture 5 ([36]). All finitely-presented groups have d-Σ2 Scott sentences. The coding technique we employed is very powerful and has other uses. Indeed, we used it to show that the class of finitely-generated groups is universal, i.e., as hard as any class of finitely- generated structures. While there are many classes of structures that are known be to universal, including graphs, lattices, rings, integral domains, 2-step nilpotent groups [38], and fields [60], ours is the first example of a class being universal among the finitely-generated structures. Theorem 6 ([35]). The class of finitely-generated groups, with finitely many constants named, is universal in the class of finitely-generated structures. Another direction I am interested in is studying notions with high complexity in group theory. We say a group A is cancelable if for any G and H, A + G ≅ A + H implies G ≅ H. This was first discussed as a criteria for a “satisfactory classification theorem” in Kaplansky’s Infinite Abelian Groups [48]. A finitely-generated abelian group is always cancelable [16, 76], but in general it is more complicated [67, 44, 19]. In [27], Fuchs and Loonstra gave a Π4 equivalent condition for a rank 1 torsion-free abelian group to be cancelable, which Arnold [5] said “are not easily characterized.” Harrison-Trainor and I showed that this Π4 condition is indeed optimal [37], so there can be no simpler characterization. We also gave some partial result in the infinite rank case.

1 The complexity hierarchy used here is the arithmetical hierarchy, and we have ∆1 ⊂ Σ1, Π1 ⊂ d-Σ1 ⊂ ∆2 ⊂ Σ2, Π2 ⊂ d-Σ2 ⊂ ⋯, with the ∆1 sets being computable, and considered to be simplest. 3 With Rossegger and San Mauro, I study the complexities of all homomorphisms between two given groups, and deﬁne the homomorphism spectrum and degree of homomorphism. This generalizes the topic of automorphism spectrum and degree of automorphism [49]. In a work in progress [41], we study which sets of Turing degrees can be realized as the homomorphism spectrum of two torsion-free abelian groups, and give analogues of theorems on automorphism spectra, including showing that every degree of homomorphism is hyperarithmetic and 0α is a degree of homomorphism for every computable ordinal α ≥ 2.

3. Word maps m For a group G, every element w ∈ Fm naturally defines a word map from w ∶ G → G by substitution and multiplication in G. These can be thought of as polynomials in groups. One object of study is their images w(Gm), especially in finite simple groups. The Ore conjecture, which was settled by Liebeck, O’Brien, Shavel, and Tiep [57], states that every element of a non- abelian finite simple group is a commutator, i.e. the commutator word is surjective. Lubotzky [58] showed that, while it is easy to see the image must contains the identity and be closed under automorphisms, this condition is also sufficient for a set in a finite simple group to be the image of some word map. With Cocke, I strengthened Lubotzky’s theorem to give more control on the word map with a given image [15], and used this to find the word explicitly in certain cases. To complement the results in finite simple groups, Cocke and I have also studied the images of word maps in nilpotent groups. We have studied when the image of a word map in a nilpotent group is closed under inverses [13]. This work was later generalized in [30]. We have also studied the size of the fibers of word maps in nilpotent groups [14]. In particular, we showed that a finite group is nilpotent if and only if every surjective word map has uniform fiber sizes. This complements the study of almost surjective words in finite simple groups by Larsen, Shalev, and Tiep[55]. One of my goal in this direction is the conjecture regarding the size of the fiber at 1 for nilpotent groups. In the work of Segal and Nikolov [63] and Abért[1], it was shown that a finite group is Sw−1(1)S solvable if and only if the quotient SGmS is bounded away from 0; and a finite group is nilpotent if Sw−1(g)S and only if the quotient SGmS is bounded away from 0 when positive. Amit conjectured an explicit bound in the case of nilpotent groups, which has been confirmed under various extra conditions [74, 56, 45]: Sw−1(1)S 1 Conjecture 7 ([62]). Let G be a nilpotent group. Then ≥ for any m-variable word SGmS SGS map w. Another direction I am working on is the application of word maps in cryptography. Recently, [59] and [46] gave a cryptographic protocol based on the multilinearity of the nested commutator words in nilpotent groups, which is similar to the Diffie-Hellman key exchange. In an ongoing work, Cocke and I generalize this protocol to other words and solvable groups, allowing more flexibility in the protocol. We are also working on an implementation of this protocol, which will allow multiple players, as well as any subset of them, to establish shared keys efficiently.

4. Language in groups The notion of formal languages first originated in linguistics, where Chomsky [10] gave the Chom- sky hierarchy of formal languages based on the expressive power of the corresponding grammar, or equivalently the complexity of an automaton that recognizes the language. This hierarchy includes the classes of regular, context-free, context-sensitive, and recursively enumerable languages. Many natural sets in group theory can be viewed as formal languages, and this perspective provides powerful applications. 4 The word problem, the set of words that represent the trivial elements in a group, is a formal language. An¯ıs¯ımov [3] showed that a group is finite if and only if its word problem is a regular language. Muller and Schupp [61] showed that G is virtually-free, i.e. has a finite-index free subgroup, if and only if its word problem is a context-free language. Holt, Reese, and Shapiro [43] showed that G admits a non-deterministic Cannon’s algorithm if and only if its word problem is growing context-sensitive in the sense of [20]. It is natural to ask if this extends to other language classes, in particular:

Question 8. Determine the class of groups whose word problems are multiple context-free.

Recently, there has been much progress toward this question. In [54], it was shown that this class is closed under amalgamations. Gilman, Kropholler, and Schleimer [29] provided various negative 2 results. In [68], Salvati showed that Z is in this class. I generalized Salvati’s work and found the n ﬁrst family of groups, Z , in this class but not a smaller class. My result was further strengthened by Salvati [69] to get a sharp multiplicity.

Theorem 9 ([40]). Any ﬁnitely-generated abelian group has multiple context-free word problem.

Given a group G generated by a finite subset S, one geometric invariant of the group is the growth rate, i.e. the rate of which the n-balls in the Cayley graph grows. It turns out that this geometric information is strongly related to the algebraic information of the group, as demonstrated in Gromov’s polynomial growth theorem [33]: The growth rate of a group is bounded by a polynomial if and only if the group has a nilpotent subgroup of finite index. It is then natural to ask when the growth rate is equal to a polynomial, which is related to the rationality of the growth series. The growth series is the formal power series whose n-th coefficient is the size of the n-ball. It turns out many groups have rational growth, i.e. the growth series is a rational function. One milestone theorem of Cannon [7] states that all hyperbolic groups have rational growth in any finite generating set. However, a theorem of Stoll [75] illustrates that the rationality can depend on the choice of generating set in general. I have been working on the rationality of growth in various metabelian groups. With Sánchez [42], m I am working on showing the rational growth of certain HNN extensions of Z , which generalizes the case of Baumslag-Solitar groups [6, 26] and a result by Sánchez and Shapiro [70]. With Choi and Pengitore [9], I am working on showing the rational growth of any 3-dimensional SOL lattice, generalizing results by Putman [65] and Parry [64]. These rationality results are often established by carefully choosing an unambiguous context-free geodesic representatives in the groups, then computing the growth series explicitly using the DSV method. This also allows us to compute the growth rate explicitly. My goal is to generalize to any metabelian groups:

Question 10. Which metabelian groups have rational growth in some generating set?

Many of the normal forms found in my work can be applied to other questions. It was conjectured by Rivin [66] that a hyperbolic group has rational conjugacy growth if and only if the group is virtually cyclic. One direction is proved by Ciobanu, Hermiller, Holt, and Rees [12], and the other by Antol´ınand Ciobanu [4]. Evetts [25] generalized this and showed that any virtually abelian group has rational conjugacy growth. In a work in progress with Ciobanu and Evetts [11], using the normal form for geodesics in Baumslag-Solitar groups, we showed that they have irrational conjugacy growth, giving evidence of Rivin’s conjecture beyond hyperbolic groups. I plan to apply my work on rational growth to the study of the conjugacy growth of metabelian groups in order to understand the following conjecture.

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