[Paper Review] The failure of the uncountable non-commutative Specker Phenomenon

Saharon Shelah, Lutz Strüngmann|ArXiv.org|Sep 5, 2000

Advanced Topology and Set Theory3 citations

TL;DR

This paper demonstrates that the non-commutative Specker Phenomenon—where homomorphisms from a countable complete free product to a group depend only on finitely many coordinates—fails in the uncountable case. By constructing $2^{2^ u}$ distinct homomorphisms from the complete free product of $ u$ non-trivial groups to $$\mathbb{Z}\u0024$ for uncountable $$\nu\u0024$, the authors show the phenomenon does not extend beyond countable index sets.

ABSTRACT

Higman proved in 1952 that every free group is non-commutatively slender, this is to say that if G is a free group and h is a homomorphism from the countable complete free product (X_omega Z) to G, then there exists a finite subset F of omega and a homomorphism h:*_{i in F} Z --> G such that h=h rho_F, where rho_F is the natural map from (X_{i in omega})Z to *_{i in F}Z . Corresponding to the abelian case this phenomenon was called the non-commutative Specker Phenomenon. In this paper we show that Higman's result fails if one passes from countable to uncountable. In particular, we show that for non-trivial groups G_alpha (alpha in lambda) and uncountable cardinal lambda there are 2^{2^lambda} homomorphisms from the complete free product of the G_alpha 's to the ring of integers.

Motivation & Objective

• To investigate whether Higman's non-commutative Specker Phenomenon, valid for countable index sets, extends to uncountable cardinals.
• To resolve Eda's open question on whether the non-commutative Specker Phenomenon holds in the uncountable setting.
• To construct explicit families of homomorphisms from uncountable complete free products to $\mathbb{Z}$ that depend on infinitely many coordinates.
• To demonstrate the maximal possible size of the homomorphism set from such products to $\mathbb{Z}$.

Proposed method

• Define the complete free product $\times\times_{\alpha \in \lambda} G_\alpha$ as the quotient of the set of all words under an equivalence relation based on finite subwords.
• Use reduced words of uncountable cofinality $\lambda$ to define homomorphisms via counting occurrences of subwords modulo isomorphism.
• Construct $2^{2^\lambda}$ almost disjoint families of reduced words $M_\alpha$ indexed by $\alpha \in 2^{2^\lambda}$, each with uncountable cofinality.
• Define homomorphisms $\varphi_\alpha$ by assigning to each word $X$ the difference between the number of positive and negative occurrences of subwords isomorphic to $M_{\beta,\gamma}$ for $\beta \in I_\alpha$, $\gamma < \lambda$, modulo $\sim_\lambda$.
• Prove that $\varphi_\alpha$ is a well-defined group homomorphism by showing compatibility under word reduction and composition.
• Establish that the $\varphi_\alpha$ are pairwise distinct due to the almost disjointness and condition (*) ensuring no overlap in subword patterns.

Experimental results

Research questions

• RQ1 Does the non-commutative Specker Phenomenon, which holds for countable complete free products, extend to uncountable index sets?
• RQ2 Can one construct uncountably many distinct homomorphisms from the complete free product of $\lambda$ non-trivial groups to $\mathbb{Z}$ when $\lambda$ is uncountable?
• RQ3 Is the size of the homomorphism set $\mathrm{Hom}(\times\times_{\alpha \in \lambda} G_\alpha, \mathbb{Z})$ maximal possible for uncountable $\lambda$?
• RQ4 What structural conditions on words in the complete free product prevent homomorphisms from factoring through finite subproducts?

Key findings

• The non-commutative Specker Phenomenon fails for uncountable index sets: there exist homomorphisms from $\times\times_{\alpha \in \lambda} G_\alpha$ to $\mathbb{Z}$ that do not factor through any finite subproduct.
• For any uncountable cardinal $\lambda$ and non-trivial groups $G_\alpha$, there are $2^{2^\lambda}$ distinct homomorphisms from the complete free product to $\mathbb{Z}$, achieving the maximum possible cardinality.
• The construction relies on a family of $2^{2^\lambda}$ almost disjoint reduced words of uncountable cofinality $\lambda$, each defining a distinct homomorphism via subword counting.
• The homomorphisms $\varphi_\alpha$ are well-defined and distinct because the underlying word families satisfy condition (*), ensuring no overlap in subword patterns across different $\alpha$.
• The uncountable cofinality of $\lambda$ is essential; the failure does not occur in the countable case due to Higman’s original result.
• The existence of such homomorphisms shows that the structure of uncountable complete free products is significantly richer than their countable counterparts.