Free structures and limiting density

We begin with Gromov’s original question, which concerned finitely presented groups. We then generalize this question to finitely presented members of a variety $V$. Section 2 has some background on algebraic varieties. Section 3 has four examples illustrating different possible behaviors and also introduces key ideas that will appear in later proofs. The first example shows that in a bijective structure with a single identity, the sentences true in the free structures have limiting density $1$. In the second example, the set of sentences with limiting density 1 is the theory of a structure in the variety, but this structure is not finitely generated, nor is it free. In the third example, we look at bijective structures as in the first example but with two identities, and we give a sentence for which the limiting density does not exist. In the fourth example, we look at abelian groups and give sentences where the limiting density is strictly between $0$ and $1$.

In Section 4, we consider more general bijective varieties. We show that for these varieties and presentations with a single generator and a single identity, a sentence has limiting density $1$ if and only if it is true in the free structure. For a language with unary function symbols $f_{1},\ldots,f_{n}$, injective and commuting, we say how to find elementary invariants.¹¹1We are grateful to Sergei Starchenko, who, having seen our result for bijective structures, suggested that we look further at varieties for languages with unary functions. We use a version of Gaifman’s Locality Theorem, which we prove using saturation.²²2We are grateful to Phokion Kolaitis for alerting us to Gaifman’s Theorem and providing references. In Section 5, we consider sentences with constants. For an arbitrary variety and presentations with a fixed tuple $\bar{a}$ of generators, we give conditions guaranteeing that, for sentences $\varphi$ in the language with added constants naming the generators, $\varphi$ has limiting density $1$ if and only if it is true in the free structure on $\bar{a}$. In Section 6, we give further examples illustrating the results from Sections 4 and 5.

Here, we recall Gromov’s original question and mention some prior work on typical or random groups. The usual language for groups has a binary operation symbol (for the group operation), a unary operation symbol (for the inverse), and a constant (for the identity). Let $T$ be the theory of groups. Recall that a group presentation consists of a tuple $\bar{a}$ of generators and a set $R$ of words $w_{i}(\bar{a})$ on these generators, called relators. In the group $G$ with presentation $\langle\bar{a}|R\rangle$, $G\models t(\bar{a})=e$ if and only if $T\cup\{w(\bar{a})=e:w(\bar{a})\in R\}\models t(\bar{a})=e$. Suppose $F$ is the free group on $\bar{a}$ and $N$ is the subgroup of $F$ consisting of the elements $t(\bar{a})$ such that $T\cup\{w(\bar{a})=e\}\models t(\bar{a})=e$. Then $G\cong F/N$.

The notion of limiting density depends not just on the variety, but also on the allowable group presentations, Ol’shanskii [18] considered presentations with $m$ generators and $k$ relators, all reduced. Kapovich and Schupp [11] considered the case where $k=1$. In the Gromov “density” model, the number of relators may vary but is bounded in terms of the length of the relators and a parameter $d$. It is important to bound the number of relators in some way; otherwise, the typical group will almost surely be trivial [18].

We consider the typical group to have property $Q$ if $Q$ has density $1$. We are particularly interested in elementary properties. For a sentence $\varphi$, the density of the property of satisfying $\varphi$ will be called simply the density of $\varphi$. The typical group, in the sense of limiting density, is also called the random group.³³3Harrison-Trainor, Khoussainov, and Turetsky [9] took a different approach and considered random structures more along the lines of the Rado graph. The typical group has some properties of free groups. Gromov introduced the property of hyperbolicity and stated that the typical group is hyperbolic. Ol’shanskii [18] showed that for presentations with $m$ generators and $k$ reduced relators, the property of being hyperbolic has limiting density $1$. Kapovich and Schupp [11] showed that for presentations with $m$ generators and $1$ reduced relator, the property that all minimal generating tuples are Nielsen equivalent has limiting density $1$. Nielsen equivalence means that one tuple can be transformed into the other by a finite sequence of simple, obviously reversible, kinds of steps.

Benjamin Fine, in conversation with the third author at the JMM in January of 2013, made an off-hand comment to the effect that in the limiting density sense, all groups look free. Fine’s comment gave rise to the conjecture below, saying that for presentations with $m\geq 2$ generators and $1$ relator, the typical group has the same elementary first order theory as the the free group. By a result of Sela [20] (see also Kharlampovich-Miasnikov [12]), the elementary first-order theory of all non-abelian free groups is the same. The conjecture is given in [10, Conjecture 2.2].

There is some evidence for the conjecture. By a result of Arzhantseva and Ol’shanskii [2, 1], a random group obtained from a presentation with $m$ generators and $k$ relators has many free subgroups. Thus, an existential sentence true in the free group is also true in a random group. Kharlampovich and Sklinos [13] used Gromov’s density model, with parameter $d$. In this setting, they showed the following.

The Kharlampovich-Sklinos result implies the conjecture (for universal sentences), but we will not give a proof here.

The term “algebraic variety” is used to mean different things in algebraic geometry and in universal algebra. The definition that we give below is the one from universal algebra.

For our purposes, it is convenient to use the following equivalent definition, of “equational class.”

Birkhoff showed that these two definitions are equivalent. Mal’tsev defined a broader class of theories whose models have well-defined presentations. See [3] for a general overview of universal algebra, where the result below appears as Theorem 11.9.

In the usual language for groups, namely $\{\cdot,^{-1},e\}$, the group axioms have the required form. Thus, groups form a variety.

Now we consider an arbitrary algebraic variety $V$. For a fixed generating tuple $\bar{a}$, there is a well-defined free structure $F$ generated by $\bar{a}$. If $\mathcal{A}$ is a structure in $V$ generated by $\bar{a}$, then $\mathcal{A}$ is a quotient of $F$ under an appropriate equivalence relation on terms $t(\bar{a})$. This equivalence relation becomes equality in $\mathcal{A}$.

We ask what the typical behavior is for members of a variety given by presentations of a specific form.

In this paper, almost all of the languages of our varieties will be either the group language or a language with just unary function symbols. We consider presentations with a fixed generating tuple $\bar{a}$, say of length $m$. For the analogue of the Ol’shanskii setting, we consider presentations with $k$ identities for some fixed $k$. For the analogue of the Kapovich-Schupp setting, we set $k=1$. This is the primary case we will consider. Where we do consider $k>1$, our presentations have the form $\bar{a}|R$, where $R$ is an unordered set of identities.

We may restrict the identities in certain natural ways. For groups, we do what the group theorists do; that is, we suppose that the identities have the form $w(\bar{a})=e$, where $w(\bar{a})$ is a word representing a product of various $a_{i}$ and $a_{i}^{-1}$. For the variety in the language consisting of two unary function symbols $S,S^{-1}$ with axioms saying that the two functions are inverses, we may restrict in a similar way, allowing only identities of the form $t(a_{i})=a_{j}$; that is, with function symbols only on the left. For the language with finitely many unary function symbols $f_{1},\ldots,f_{n}$ and varieties that do not have axioms explicitly saying that one $f_{j}$ is the inverse of another $f_{i}$, our identities have the form $t(a_{i})=t^{\prime}(a_{j})$, where $t(x)$ and $t^{\prime}(x)$ are terms built up from the function symbols.

Before we begin, we state three lemmas that hold very generally.

We start with the variety of bijective structures. Recall that the language consists of two unary function symbols $S,S^{-1}$. The axioms are

These guarantee that the function $S$ is $1-1$ and onto and that $S^{-1}$ is the inverse of $S$. Let $T$ be the theory generated by these axioms. The models consist of infinite $\mathbb{Z}$-chains and finite cycles $\mathbb{Z}_{m}$. While these structures lack the mathematical interest and importance of groups, it is instructive to consider them because there are relatively simple elementary invariants, and for presentations with a single generator and a single identity, we can calculate the limiting densities for these sentences. It turns out that the analogue of Conjecture 1.2 holds.

We consider bijective structures with a single generator $a$. There is a single $\sim$-class, which has the form $\mathbb{Z}$, an infinite chain, or $\mathbb{Z}_{m}$, a cycle of size $m$. We note that in either $\mathbb{Z}$ or $\mathbb{Z}_{m}$, all elements are automorphic. The following lemma is clear from the meanings of the sentences $\alpha(n,k)$ and $\beta(n)$.

For models with a single generator, $\alpha(n,k)$ is false for $k>1$, and $\beta(n)$ is equivalent to $\bigwedge_{m\leq n}\neg{\alpha(n,1)}$. Thus, it is enough to consider the elementary invariants of the form $\alpha(n,1)$.

Here our presentations have a single identity, of the form $t(a)=a$ (function symbols occur only on the left). We may refer to the term $t(a)$ as a relator. For a single relator $t(a)$, we get $\mathbb{Z}$ if for some $k$, $t(a)$ has $k$ occurrences of $S$ and $k$ occurrences of $S^{-1}$. We get $\mathbb{Z}_{m}$ if for some $k$, $t(a)$ has either $m+k$ occurrences of $S$ and $k$ of $S^{-1}$ or $m+k$ occurrences of $S^{-1}$ and $k$ of $S$.

We will show that for all $n\geq 1$, $\alpha(n,1)$ has limiting density $0$. This implies that $\neg\alpha(n,1)$, which is true in the free structure, has limiting density $1$. We will use two combinatorial lemmas. The first is an approximation for $\left(\begin{array}[]{c}2k\\ k\end{array}\right)$, good for large $k$. The proof requires the use of Stirling’s formula on all three factorials (see the website of Das [5]).

The second combinatorial lemma is an inequality.

To show that $\alpha(n,1)$ has limiting density $0$, we use several further lemmas.

The next lemma bounds the sum $P_{n+2k}(\alpha(n,1))$ by a single term.

Now we can show that the limiting density of $\alpha(n,1)$ is $0$ for any $n\geq 1$. To do so, we must make an odd/even case distinction because the only way to get presentations of different lengths of the same structure is for these lengths to differ by a multiple of two, so $P_{n+2k+1}(\alpha(n,1))$ will equal $P_{n+2k}(\alpha(n,1))$. However, if $s=n+2k$ for some $k$, then $P_{s}(\alpha(n,1))$ has a new last term $\left(\begin{array}[]{c}n+2k\\ k\end{array}\right)$, and $P_{s+1}(\alpha(n,1))=P_{s}(\alpha(n,1))$. Therefore, it is enough to show that

By Lemma 3.4, the last term of $P_{n+2k}(\alpha(n,1))$ is greater than the sum of the earlier terms. Thus, $P_{n+2k}(\alpha(n,1))<2\left(\begin{array}[]{c}n+2k\\ k\end{array}\right)$. Recall that $P_{n+2k}=2^{n+2k+1}-1$, which is strictly greater than $2^{n+2k}$, so $\frac{P_{n+2k}(\alpha(n,1))}{P_{n+2k}}<\frac{2}{2^{n+2k}}\left(\begin{array}[]{c}n+2k\\ k\end{array}\right)$. To prove that the limiting density of $\alpha(n,1)$ is $0$, it is enough to prove that

We can express $\frac{2}{2^{n+2k}}\left(\begin{array}[]{c}n+2k\\ k\end{array}\right)$ as a product of two factors, one involving the fixed $n$ and the other not. The first factor is

This is an $(n+1)$-fold product with limit $2$ as $k\rightarrow\infty$. The second factor is $\frac{1}{2^{2k}}\left(\begin{array}[]{c}2k\\ k\end{array}\right)$. By Lemma 3.3 above, this is $(1+o(1))\frac{1}{\sqrt{\pi k}}$, which has limit $0$. All together, we have shown the following.

From this, we get the following.

For later use, we state below another immediate consequence of Proposition 3.8. For a term $t(a)$, let $X$ be the difference between the number of occurrences of $S$ and the number of occurrences of $S^{-1}$ in $t(a)$. The relators that make $\alpha(n,1)$ true are exactly those for which $|X|=n$.

Next, take the language with a single unary function symbol $f$, and the variety with no axioms. We consider presentations with a single generator $a$ and a single identity. At first, we focus on the sentence $\varphi$ saying that $f$ is not $1-1$. We will see that this sentence is false in the free structure, but it has limiting density $1$. After that, we will consider arbitrary sentences, and prove a zero-one law.

For an identity $f^{(r)}(a)=f^{(r^{\prime})}(a)$, where $r,r^{\prime}\geq 0$, the length is $r+r^{\prime}$. If $r=r^{\prime}$, then we get an $\omega$-chain. If $0<r<r^{\prime}$, then we get a chain of length $r$ leading to a cycle of length $r^{\prime}-r$. If $0=r<r^{\prime}$, then we get a cycle of length $r^{\prime}$. Similarly, if $0<r^{\prime}<r$, we get a chain of length $r^{\prime}$ leading to a cycle of length $r-r^{\prime}$, and if $0=r^{\prime}<r$, we get a cycle of length $r$.

We write $m+\mathbb{Z}_{n}$ for a chain of length $m$ leading to a cycle of length $n$, allowing the possibility that $m=0$. Any structure in our variety obtained from a single generator and a single identity has one of the following forms:

1. 1.

   an $\omega$-chain—this is the free structure,

2. 2.

   a finite chain leading to a finite cycle, or

3. 3.

   a finite cycle.

Recall that $\varphi$ holds if the function is not $1-1$. We can see that $\varphi$ is false in $\omega$ and the finite cycles $\mathbb{Z}_{n}$, and true in the structures $m+\mathbb{Z}_{n}$ for $m\geq 1$, so an identity $f^{n}(a)=f^{n^{\prime}}(a)$ yields a structure in which $\varphi$ is false if $n=n^{\prime}$ or if one of $n,n^{\prime}$ is $0$, and true otherwise. We can easily count the identities of length $m$ that make $\varphi$ false. For $m=0$, there is just one identity, and it makes $\varphi$ false. For $m=1$, there are two identities, and both make $\varphi$ false. For $m\geq 2$, if $m$ is odd, there are just two identities that make $\varphi$ false, and if $m$ is even, there are three identities that make $\varphi$ false. Thus, $P_{s}(\neg{\varphi})=1+2+3+2+\ldots+(\frac{5}{2}+\frac{1}{2}(-1)^{s})$.

In fact, we will see that each sentence has limiting density $1$ or $0$. The set of sentences with limiting density $1$ is not that of the free structure. It is the theory of a structure $\mathcal{A}$ that we may think of as a limit of the structures $m+\mathbb{Z}_{n}$ (see Figure 1). The limit structure $\mathcal{A}$ consists of an $\omega$-chain together with two $\omega^{*}$-chains and a single $\omega$-chain that come together at a special point—this point is the end of the two $\omega^{*}$-chains and the beginning of the $\omega$-chain (see Figure 2). The chain of length $m$ is replaced, in the limit, by an $\omega$-chain plus one of the $\omega^{*}$-chains, and the $n$-cycle is replaced, in the limit, by the other $\omega$-chain and $\omega^{*}$-chain. We note that $\mathcal{A}$ is not finitely generated.

We conclude the following.

In the next example, we return to the variety of bijective structures as in Section 3.1. As before, our presentations have a single generator $a$, but there are two identities instead of just one. We will show that the limiting density need not exist. Recall that the language consists of unary function symbols $S,S^{-1}$ and that the axioms say that $S$ and $S^{-1}$ are inverses. The identities have the form $t(a)=a$, with function symbols only on the left. Each identity is thus equivalent to one of the form $S^{m}(a)=a$, where $m\in\mathbb{Z}$.

The proof is somewhat involved. We begin with some elementary lemmas, but eventually we will consider a random walk on a group and appeal to results from random group theory that depend on the Central Limit Theorem. The lemma below tells us when the sentence $\varphi$ is true.

Our presentations have two identities, but we also need some facts about single identities. We indicate with ^′ that we are considering single identities, writing $P^{\prime}_{=m}$ for the number of identities of length $m$ and $P^{\prime}_{s}$ for the number of length at most $s$, and writing $P^{\prime}_{=m}(B)$, $P^{\prime}_{s}(B)$ for the number of these identities in a set $B$. We reserve $P_{s}$ for the number of unordered pairs of identities of length at most $s$, and we write $P_{s}(B^{2})$ for the number such that both identities are in $B$.

For a single identity of the form $t(a)=a$, let $X$ be the difference between the number of occurrences of $S$ and the number of occurrences of $S^{-1}$ in $t$. Intuition may suggest that the statement $n|X$ should have limiting density $\frac{1}{n}$. This turns out to be true for odd $n$. However, for $n=2$, we find that the limiting density for the statement $2|X$ does not exist. Essentially, the reason is that the last term of $P^{\prime}_{s}(2|X)=\sum_{m\leq s}P^{\prime}_{=m}(2|X)$ may be greater than the sum of all earlier terms, and this term depends on the parity of $s$. The lemma below says what happens to $P_{=s}(2|X)$ as the parity of $s$ changes.

The next lemma gives the proportion of single identities of length at most $s$ for which $2|X$ holds. The value depends on the parity of $s$.

So far, the lemmas have involved only elementary calculations. The next result is from random group theory [4, 19], concerning a random walk on a group. The elements of the group $G=\mathbb{Z}_{n}$ represent the possible remainders after division of an integer $z$ by $n$. In general, for a random walk, there are finitely many states, and given just the current state $s$, with no more prior history, we have fixed probabilities of passing next to state $s^{\prime}$. We allow $s^{\prime}=s$.