REIDEMEISTER CLASSES IN SOME WREATH PRODUCTS BY Zk

The paper proves two results: (i) existence of automorphisms with finite Reidemeister number for G ≀ Z^k in three explicit families (Theorem 3.1), and (ii) for any finite-order automorphism ϕ, if R(ϕ) < ∞ then TBFTf holds (Theorem 4.1 and Corollary 4.2). The proof strategy hinges on the extension-theoretic summation formula (Theorem 2.2) and on explicit blockwise constructions that make Id − (α(t) ∘ ϕ′) onto in each summand, together with R(ϕ̄) = |det(I − A)| for ϕ̄ on Z^k (formula (4)) . In Case 1, the paper crucially mixes copies using 2×2 and 3×3 blocks (F2, F3) and checks invertibility in the ‘exceptional case’ as well, thereby handling odd multiplicities for the 3-primary part; the identities and explicit inverses are written out and determinant checks are done mod 2 and mod 3 (see (5)–(6) and the block inverses) . For Case 2, the order-3 matrix M satisfies M^2 + M + I = 0 (eq. (7)), which ensures finite affine orbits and gives R(ϕ̄) = 3^{k/2} . For Case 3, an order-5 block in GL_4(Z) is used analogously. The TBFTf part follows because for finite-order ϕ one has R(ϕ′) ∈ {1, ∞}, and when R(ϕ) < ∞ all summands equal 1; separability by a finite quotient then yields TBFTf . By contrast, the candidate solution follows the same high-level plan (choose A of order 2, 3, or 5; use the summation formula; force Id − θ^m onto for orbit lengths m) but it fails to treat odd multiplicities in the 3-primary component in Case 1. On a single Z/3^rZ summand there is no unit u with both 1 − u and 1 − u^2 invertible, so one cannot achieve Id − θ and Id − θ^2 both surjective in dimension 1; this is precisely why the paper introduces a 3×3 block F3 in Case 1. The model proposes only 2×2 blocks for p = 3 and does not discuss the leftover 1-dimensional summand when d_i is odd, leading to a gap. The TBFTf portion of the model agrees with Theorem 4.1 and Corollary 4.2 of the paper .