2. Background

We begin by defining the k-gonal model of random groups.

Definition 2.1 ([Reference Ashcroft and Roney-Dougal1]). Fix a set S of cardinality n. The k-gonal model of random groups, denoted $\mathcal{M}_k(n, d),$ is the set of group presentations $\langle S|R\rangle$ such that R is a set of $(2n-1)^{kd}$ cyclically reduced words on $S\cup S^{-1}$ of length k, along with the uniform probability measure. We say that a k-gonal random group at density d satisfies a property P with overwhelming probability, abbreviated w.o.p, if

\begin{equation*} \lim_{n\to\infty}\mathbb{P}(\mbox{the group defined by a presentation in }\mathcal{M}_k(n, d) \mbox{ satisfies P}) = 1. \end{equation*}

A related family of models of random groups restricts possible relators to positive words, i.e. words on S but not on $S^{-1}$ . These are called the positive k-gonal models of random groups and are denoted $\mathcal{M}^+_k(n, d).$ We will relax the condition on the number of relators in this model and require that the number of relators be $ f(n) \approx (2n-1)^{kd}$ , where “ $\approx$ ” means that $\lim_{n\to \infty} f(n)/(2n-1)^{kd}$ is a finite constant. The definition of satisfying property P w.o.p. will not depend on this detail.

We now define (Kazhdan’s) Property (T) for discrete, finitely generated groups. The notation is suggestive—the idea of Property (T) is that the trivial representation of the group must be isolated in some metric. More formally, we can define this property via (almost) invariant vectors: Let $\Gamma$ be a finite generated group with finitely generating set S. If $\mathcal{H}$ is a Hilbert space and $\pi\,:\, \Gamma \to U(\mathcal{H})$ is a unitary representation of $\Gamma$ on $\mathcal{H}$ , then $\pi$ has almost invariant vectors if, for every $\varepsilon>0$ , there exists some $u_\varepsilon \in \mathcal{H}$ such that for every $s \in S$ , $\|\pi(s)u_\varepsilon - u_\varepsilon\| < \varepsilon\|u_\varepsilon\|.$ A vector $u \in \mathcal{H}$ is invariant if $\pi(g)(u) = u$ for every $g \in \Gamma$ .

The first results towards demonstrating Property (T) in random groups were first stated in [Reference Żuk14] and fully proved in [Reference Kotowski and Kotowski5]. The proof involved passing between several different models of random groups and analyzing spectral properties of random graphs.

In fact, they also showed that this is true in $\mathcal{M}_3(n, d)$ and in the Gromov model of random groups (at the same density).

The tempting argument is to say if $G \in \mathcal{M}_{jk}(n, d)$ then $G = \langle S | R \rangle$ and any relator in R can be interpreted as a word of length k over the alphabet of words of length j on $S \cup S^{-1}$ , denoted $|W_j|$ . Then, G is in $\mathcal{M}_k(|W_j|, d).$ However, as is pointed out in [Reference Kotowski and Kotowski5], problems arise around the requirement that relators be reduced. In particular, a word of length kj is a word of length k in the alphabet of words of length j; however, a reduced word of length k in the alphabet of words of length j might not be reduced as a word of length jk. So we must first consider an intermediate group, $G^+$ , whose relators are positive words on S. This removes the reduction problem, and we can safely interpret this group as a random group in $\mathcal{M}_k^+(n, d)$ .

The method of proof closely follows the argument used to prove Theorem 4.3 in [Reference Odrzygozdz9], which shows that groups in the 6-gonal model w.o.p. satisfy Property (T). In particular, to prove that a random group $G \in \mathcal{M}_{jk}(n,d)$ satisfies Property (T), we construct two groups $G^+$ and $\Gamma$ so that

\begin{equation*} \Gamma \stackrel{\phi}{\twoheadrightarrow} \phi(\Gamma) \stackrel{f.i.}{\leq} G^+ \twoheadrightarrow G,\end{equation*}

and then show that $\Gamma$ satisfies Property (T).

There are two parts to this proof. The first is showing that $\phi(\Gamma)$ is finite index in $G^+$ . We find a finite set of coset representatives to prove this explicitly, generalizing the method used in [Reference Odrzygozdz9]. The second is in showing that the group $\Gamma$ can be interpreted as a typical representative of random groups in $\mathcal{M}^+_k(m, d^{\prime})$ for $d^{\prime} >1/3.$ For this part of the argument, we mimic the method developed in the proof of Theorem A in [Reference Kotowski and Kotowski5].

3. Property (T) in jk-gonal models

Fix a generating set S of cardinality n. Let $W^+$ be the set of positive words on S and let $W_j$ be the words of length j on S. For any set R of words on S, let $R^+ = R \cap W^+$ . Then for any group $G = \langle S | R\rangle$ in $\mathcal{M}_{jk}(n, d)$ , there is a group $G^+ = \langle S | R^+\rangle$ that surjects onto G.

Since G is in the jk-gonal model, every word in R and $R^+$ has length jk. There is a natural injective map $\phi: W^+_j \to F(S)$ , where F(S) denotes the free group on S, given by the labeling on each word in $W^+_j$ . Define $R^+_k = \phi^{-1}(R^+)$ and notice that every word in $R^+_k$ has length k over $W_j^+$ . Let $\Gamma = \langle W_j^+ | R^+_k \rangle$ . Then, $\phi$ induces a homomorphism, also denoted $\phi$ , mapping $\Gamma \to G^+$ .

Note also that $|R^+_k| = |\phi(R^+_k)| = |R^+|,$ so if $G^+ \in \mathcal{M}^+_{jk}(n, d^{\prime}),$ then $\Gamma \in \mathcal{M}^+_k(|W^+_j|, d^{\prime})$ . Furthermore, this map takes the uniform probability distribution on $\mathcal{M}^+_{jk}(n, d^{\prime})$ to the uniform probability distribution on $\mathcal{M}^+_k(|W^+_j|, d^{\prime})$ , because $\phi$ induces a 1-1 map on possible sets of relators.

We will show that $\phi(\Gamma)$ is a finite index subgroup of $G^+$ by finding an explicit set of coset representatives of $\phi(\Gamma)$ in $G^+$ . We note that an element of $G^+$ is in $\phi(\Gamma)$ if and only if it is expressible as a string of words of length k, all of which are positive or the inverse thereof. We do this by inserting trivial words that allow us to group positive and negative letters into subwords of length j.

In the following proof, for any word w, $|w|_j$ denotes the word length of w modulo j.

Proof. Let w be a word on $S \cup S^{-1}$ . Without loss of generality, assume that the first letter of w is in $S^{-1}$ . Then, w can be written as

\begin{equation*}w = A_\ell a_\ell \cdots A_2 a_2 A_1 a_1,\end{equation*}

where $\{a_i\}$ are positive words on S, $\{A_i\}$ are negative words on S, and $a_1$ might be empty.

Let x be a positive word of length $j - |a_1|_j$ and let y be a positive word of length $j - |A_1x^{-1}|_j$ . Then, we have

\begin{equation*}w = A_\ell a_\ell \cdots A_2 a_2 y y^{-1}A_1 x^{-1}xa_1.\end{equation*}

Note that $xa_1$ is a positive word with length divisible by j, and $y^{-1}A_1 x^{-1}$ is a negative word with length divisible by j. Therefore, $w = A_\ell a_\ell \cdots A_2 a^{\prime}_2 u,$ where $u \in \phi(\Gamma)$ . By induction, we see that $w \in A^{\prime}_\ell\phi(\Gamma)$ where $A^{\prime}_\ell$ is a positive word on S. Then, $A^{\prime}_\ell = a z,$ where j divides $|z|$ and $|a|< j$ , so $z \in \phi(\Gamma)$ . Therefore, $w = azv$ , where $zv \in \phi(\Gamma)$ and $|a| < j$ .

Thus, the (finite) set of words of length $< j$ contains a transversal of $\phi(\Gamma)$ in $G^+$ , and $\phi(\Gamma)$ has finite index in $G^+$ .

We are now ready to prove Theorem 1.1:

Proof. We first show that a random group $G^+$ in $\mathcal{M}^+_{kj}(n, d)$ has Property (T). Let $m = |W^+_{j}|$ . By the construction above, given such a $G^+$ we can find a group $\Gamma$ in $\mathcal{M}^+_k(m, d)$ and a map $\phi: \Gamma \to G^+$ . The distribution induced on $\mathcal{M}^+_k(m, d)$ in this way is uniform and by Lemma 3.1 $\phi(\Gamma)$ is a finite index subgroup of $G^+$ . Since $\Gamma$ satisfies Property (T) with overwhelming probability, by Remark 2.4, so does $G^+$ .

Now let $G =\langle S|R\rangle$ be a group in $\mathcal{M}_{kj}(n, d).$ Note that $|W^+_j| > \frac{1}{2^j}|W_j|.$ Therefore, the expected number of elements of $R^+$ is at least $\frac{1}{2^j}|R|> \frac{1}{2^j}|R| = \frac{1}{2^j}(2n-1)^{kjd} > 2(2n-1)^{kjd^{\prime}}$ , for any $d^{\prime}<d$ and sufficiently large n. The Chernoff bound then implies that with overwhelming probability, $|R^+| > (2n-1)^{kjd^{\prime}}.$

Fix $d^{\prime} \in (d_0, d)$ . If we enumerate the elements of $W^+_j$ and take R ^′ to be the set of the first $(2n-1)^{kjd^{\prime}}$ elements of $W^+_j$ which are in R, we can find a group $G^+ \langle S | R^{\prime}\rangle\in \mathcal{M}^+_{kj}(n, d^{\prime})$ that surjects onto G. This gives a uniform distribution on $\mathcal{M}^+_{kj}(n, d^{\prime})$ , so w.o.p $G^+$ satisfies Property (T), and by Remark 2.4 so does G.

By Theorem 3.14 in [Reference Kotowski and Kotowski5], with overwhelming probability random groups in $\mathcal{M}^+_3(n, d)$ satisfy Property (T) at density $d>1/3$ . By applying Theorem 1, we get the following result.