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Between free and direct products of groups

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Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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Maxime Gheysens, Nicolas Monod

Group Theory · Submitted 10 Jan 2022

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim This construction curiously shares some properties with the free product but others with the direct product.

how it is stated · asserted 0.98 read the record →

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arXiv ID2201.03625 (version 1)
Pith IDYD4FWY6P, Pith's permanent record of this paper and its review
CitationMaxime Gheysens, Nicolas Monod. "Between free and direct products of groups." arXiv:2201.03625 (2022). https://pith.science/paper/YD4FWY6P
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,917 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.09 · human participants 0.02 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.45 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We investigate the group $G \vee H$ obtained by gluing together two groups $G$ and $H$ at the neutral element. This construction curiously shares some properties with the free product but others with the direct product. Our results address among others Property (T), CAT(0) cubical complexes, local embeddability, amenable actions, and the algebraic structure of $G \vee H$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We investigate the group $G \vee H$ obtained by gluing together two groups $G$ and $H$ at the neutral element. contribution 0.91 0.70 0.04 asserted 0.94
s1 This construction curiously shares some properties with the free product but others with the direct product. background 0.50 0.21 0.09 asserted 0.98
s2 Our results address among others Property (T), CAT(0) cubical complexes, local embeddability, amenable actions, and the algebraic structure of $G \vee H$. contribution 0.85 0.90 0.17 asserted 0.98

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We investigate the group $G \vee H$ obtained by gluing together two groups $G$ and $H$ at the neutral element. This construction curiously shares some properties with the free product but others with the direct product. Our results address among others Property (T), CAT(0) cubical complexes, local embeddability, amenable actions, and the algebraic structure of $G \vee H$.

Human review

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How the literature uses it

2 arXiv papers cite this in 5 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 1 sentence without a confident label

Cites it as background 2 papers
  • Recall from [ref] the definition of the $⋁$-product.

    Lamplighter-like geometry of groups · Halos of groups · 0.98

  • These groups are referred to as lampshuffler groups[The name “lampshuffler” seems to have first appeared in [ref] , and it is used there to refer to the group $FAlt(H)⋊H$, where $FAlt(H)$ is the group of finitary even permutations of $H$.

    The Poisson boundary of lampshuffler groups · Introduction · 0.99

What it cites · 7 sentences without a confident label

Uses the method or tool of 1 paper
  • The proof begins with an argument that we borrow from [ref] ; after that, we let $G$ and $H$ compete for the insufficient space in $G ∪H$.

    Near actions · The extension usually does not split · 1.00

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