LEF Growth of Wreath Products with Abelian and Virtually Abelian Lamps

Proof.

The lower bound follows from word growth. Choose \(0\neq a\in A\), and let

be a geodesic segment in \(\Z^d\), with \(L=\lfloor cn\rfloor\). For each subset \(S\subseteq\{0,\ldots,L\}\), put the lamp value \(a\) at the positions \(x_i\) with \(i\in S\), put the lamp value \(0\) at the remaining positions, and end the cursor at \(x_L\). If \(c>0\) is small enough, all these elements have word length at most \(n\). They are pairwise distinct, so the \(n\)-ball contains at least \(2^{L+1}\) elements. Hence

For the upper bound, write elements as \((P,v)\), with

The ball \(B_{A\wr\Z^d}(n)\) has at most exponentially many elements. Moreover, for all elements in this ball, the exponents occurring in \(P\) lie in \([-Cn,Cn]^d\), and the coefficients in the free part of \(A\) have norm at most \(Cn\).

Let \(\mathcal P_n\) consist of the following non-zero Laurent polynomials:

1. all differences \(P-P'\), where \((P,v),(P',v)\in B_{A\wr\Z^d}(n)\) have the same base component and \(P\neq P'\);

2. all polynomials \(a_0(t^u-1)\), where \(a_0\in A\) is a fixed non-zero element and \(0\neq u\in\Z^d\) satisfies \(\|u\|_1\le 2n\).

The first family has cardinality at most \(\exp(Cn)\), because the ball has exponential size. The second family has polynomial cardinality. Hence Lemma 4.1 applies to \(\mathcal P_n\). We obtain \(V\), commuting automorphisms \(T_1,\ldots,T_d\), and an equivariant evaluation map \(\Theta\), with

Define

This is a homomorphism because \(\Theta(t^vP)=T^v\Theta(P)\). We claim that \(\Phi\) is injective on \(B_{A\wr\Z^d}(n)\).

Suppose

for two elements in the \(n\)-ball. Then \(T^{v-v'}=1\). If \(v\neq v'\), then \(0\neq u=v-v'\) has \(\|u\|_1\le 2n\), and

contradicting the construction of \(\mathcal P_n\). Therefore \(v=v'\). Then

and the first family in \(\mathcal P_n\) forces \(P=P'\). Thus \(\Phi\) locally embeds the \(n\)-ball into a finite group of order at most \(\exp(Cn)\), proving the upper bound.

Proof.

If \(\Delta\) is abelian, this is Theorem 4.2.

Assume \(\Delta\) is non-abelian. The lower bound follows from Proposition 3.1, since

For the upper bound, choose \(M\simeq n\) large enough that the quotient map

is injective on the base ball and support region needed to multiply all elements of \(B_{\Delta\wr\Z^d}(n)\). Equivalently, apply Lemma 3.3, with \(\eta=\operatorname{id}_\Delta\), to the regular action of \(\Z^d\) on \((\Z/M\Z)^d\). This gives a local embedding into

Its order is

Thus \(\LEF_{\Delta\wr\Z^d}(n)\preceq\exp(n^d)\).

Proof.

If every element of the form \(p(a)-a\), with \(a\in A\) and \(p\in P\), were torsion, then \([A,P]\) would be contained in the torsion subgroup of the finitely generated abelian group \(A\), and hence would have rank \(0\). Since \(\rho>0\), this is impossible. Therefore there exist \(b\in A\) and \(\tau\in P\) such that \(\tau(b)-b\) has infinite order.

Proof.

Choose \(b\in A\), \(\tau\in P\), and \(c=\tau(b)-b\) as in Lemma 6.1. Let

be a local embedding into a finite group. Choose \(\varepsilon>0\) small enough that, for every \(x\in X=B_{\Z^d}(\varepsilon n)\), every prefix of the words

lies in \(B_{\Delta\wr\Z^d}(n)\), and that all commutator words between the generators \(b_x,\tau_x\) and \(b_y,\tau_y\), for distinct \(x,y\in X\), are visible prefix-by-prefix. This is possible by Lemma 2.4, after decreasing \(\varepsilon\). We have \(|X|\simeq n^d\).

For \(x\in X\), write

Since \(A\) is written additively and \(c=\tau(b)-b\), we have, for every \(\ell\ge 0\),

Equivalently, in multiplicative notation inside the lamp group,

The prefix visibility chosen above gives, in \(Q\),

Because \(c\) has infinite order, the elements

are distinct in the ball. Hence

are distinct in \(Q\).

Let

We claim that \(|D_x/Z(D_x)|\ge c_0n\) for some constant \(c_0>0\). If the image of \(B_x\) in \(D_x/Z(D_x)\) had order \(\ell\le \varepsilon n\), then \(B_x^\ell\in Z(D_x)\). In particular, \(B_x^\ell\) would commute with \(T_x\), so

contradicting the distinctness above. Thus \(|D_x/Z(D_x)|\ge c_0n\), after changing constants.

For distinct \(x,y\in X\), the corresponding lamp subgroups commute in the wreath product, and the relevant commutator words are visible. Hence

in \(Q\). Enumerate \(X=\{x_1,\ldots,x_N\}\), with \(N\simeq n^d\), and set \(P_j=D_{x_1}\cdots D_{x_j}\). As before,

so

Therefore

which proves the proposition.

Proof.

Let \(T=\tors(A)\), and let \(e\) be the exponent of \(T\) if \(T\neq 0\), and \(e=1\) otherwise. Choose integers

with \(e\mid m\). Put

and let \(\eta:\Delta\to\Delta_m\) be the quotient homomorphism. Choose \(m\) with a sufficiently large implicit constant so that \(\eta\) is injective on all lamp values visible in \(B_{\Delta\wr\Z^d}(n)\). On the free part of \(A\) this follows from \(m\gg n\), and on the torsion part from \(e\mid m\).

Let \(\Z^d\) act regularly on

Choose \(M\simeq n\) with a sufficiently large implicit constant so that the pointed Schreier ball of this action agrees with the Cayley ball of \(\Z^d\) throughout the base region needed to multiply elements of \(B_{\Delta\wr\Z^d}(n)\). Applying Lemma 3.3 with \(F=\Delta_m\) gives a local embedding of \(B_{\Delta\wr\Z^d}(n)\) into

This is a local finite-action model, not a global quotient map of the wreath product; the choice of \(M\) prevents collisions only on the visible support region.

If \(r=\rank_\Z A\), then \(|A_m|\le Cm^r\), so \(|\Delta_m|\le C'm^r\). Since \(|Y|=M^d\), the model has order

Therefore

This proves the upper bound.

Proof.

Let \(T=\tors(A)\). Since \(\rho=0\), the action of \(P\) on \(A/T\) is trivial. Equivalently, for every \(p\in P\) and \(a\in A\), the element \(p(a)-a\) belongs to \(T\). Let \(e\) be the exponent of \(T\), with \(e=1\) if \(T=0\). Then

for all \(p\in P\) and \(a\in A\). Thus

is a torsion-free finite-index subgroup of \(A\), and it is centralized by \(P\). Hence \(B\) is central in \(\Delta\). The quotient

is finite.

We next construct a homomorphism from \(\Delta\) to a free abelian group that is injective on \(B\). Since \([A,P]\le T\) is finite and \(P\) is finite, the commutator subgroup \([\Delta,\Delta]\), generated by \([A,P]\) and commutators in \(P\), is finite. Let

and let \(\chi:\Delta\to A_0\) be the natural homomorphism. If \(b\in B\) maps to zero in \(A_0\), then the image of \(b\) in \(\Delta_{\mathrm{ab}}\) is torsion. Thus \(b^k\in[\Delta,\Delta]\) for some \(k\ge 1\). But \([\Delta,\Delta]\) is finite and \(B\) is torsion-free, so \(b=1\). Therefore \(\chi|_B\) is injective.

Let \(q:\Delta\to E=\Delta/B\) be the quotient map. Define

If \(\iota(g)=(1,1)\), then \(q(g)=1\), so \(g\in B\), and \(\chi(g)=1\). Since \(\chi|_B\) is injective, \(g=1\). Hence \(\iota\) is injective.

Finally, if \(\Delta\) is non-abelian, then \([\Delta,\Delta]\neq 1\). Since \([\Delta,\Delta]\) is finite and \(B\) is torsion-free, \([\Delta,\Delta]\cap B=1\). Thus some non-trivial commutator survives in \(E=\Delta/B\), so \(E\) is non-abelian.

Proof.

The lower bound follows from Proposition 3.1, since \(\gamma_{\Z^d}(n)\simeq n^d\).

For the upper bound, use Lemma 6.4 to embed

where \(A_0\) is finitely generated free abelian and \(E\) is finite. This induces an embedding

Because \(A_0\times E\) is a direct product, there is an injective homomorphism

given by projecting the lamp component to \(A_0\) and \(E\), and sending the base element diagonally to the two base components.

Theorem 4.2 gives

For the finite lamp \(E\), the torus model used in Theorem 5.1 gives the upper bound

regardless of whether \(E\) is abelian. Taking the product of the two local finite models gives

Since \(\Delta\wr\Z^d\) embeds in this group, the same upper bound holds for \(\Delta\wr\Z^d\).

Proof.

If \(\Delta\) is abelian, then \(\Delta\) is a non-trivial finitely generated abelian group, so the result is Theorem 4.2. If \(\rho=0\) and \(\Delta\) is non-abelian, the result is Proposition 6.5. If \(\rho>0\), combine Propositions 6.2 and 6.3.

Proof.

We have \(D_\infty=\Z\rtimes C_2\), where \(C_2\) acts on \(\Z\) by multiplication by \(-1\). Hence

has rank \(1\), so \(\rho>0\). The result follows from Theorem 6.6.

Proof.

The horizontal coordinates change by at most one under multiplication by \(x^{\pm1}\) or \(y^{\pm1}\), so \(|a|,|b|\le n\) in the \(n\)-ball. The central coordinate records signed area swept by a word of length \(n\), hence is \(O(n^2)\). This gives the first inclusion and the upper growth bound \(O(n^4)\).

Conversely, all elements \(x^ay^b\) with \(|a|,|b|\le cn\) have length \(O(n)\). Also, commutators satisfy

Every integer \(|c|\le cn^2\) can be written as a sum of \(O(1)\) products \(r_is_i\) with \(|r_i|,|s_i|\le C\sqrt{|c|}\le C'n\). Thus \(z^c\) has length \(O(n)\) for \(|c|\le cn^2\). This proves the second inclusion and the lower growth bound \(\Omega(n^4)\).

Proof.

In coordinates \((a,b,c)\) corresponding to \(x^ay^bz^c\), multiplication has the form

where \(\omega\) is an integral bilinear form. If \(a,b,a',b'\) are divisible by \(m\), then \(\omega((a,b),(a',b'))\) is divisible by \(m^2\). Thus \(K_m\) is a subgroup.

Every left coset of \(K_m\) has a representative \(x^\alpha y^\beta z^\gamma\), with

Let \(Y_m=K_m\backslash H\), and let \(P_m\) be the image of \(H\) in its right action on \(Y_m\).

8 A congruence lemma for abelian lamps over the Heisenberg group

The methods above do not determine \(\LEF_{C_p\wr H_3(\Z)}(n)\). The obstruction is that, for abelian lamps, one must compress sparse elements of the group algebra \(\F_p[H_3(\Z)]\), and the base is non-abelian.

We record a useful congruence lemma. It separates one sparse group-algebra element in a congruence quotient of modulus \(O(n)\), but it does not supply a single small finite module separating all visible elements simultaneously.

9 Open problems

The preceding results leave several natural cases open.

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During the preparation of this work, ChatGPT, by OpenAI, was used to assist with mathematical drafting, formalization, review, and editing. This work is shared as a preliminary AI-assisted mathematical note. The mathematical content may have been only partially reviewed and may contain errors; it should not be treated as peer-reviewed or as a fully verified manuscript.