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Hyperbolic groups with logarithmic separation profile

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Nir Lazarovich, Corentin Le Coz

Group Theory · Submitted 26 Oct 2021

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

The authors' main claim We prove that hyperbolic groups with logarithmic separation profiles split over cyclic groups.

how it is stated · asserted 1.00 read the record →

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arXiv ID2110.13595 (version 1)
Pith IDKHGLRVOZ, Pith's permanent record of this paper and its review
CitationCorentin Le Coz, Nir Lazarovich. "Hyperbolic groups with logarithmic separation profile." arXiv:2110.13595 (2021). https://pith.science/paper/KHGLRVOZ
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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, 2,933 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.88 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.23 · compares with prior results 0.44 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

We prove that hyperbolic groups with logarithmic separation profiles split over cyclic groups. This shows that such groups can be inductively built from Fuchsian groups and free groups by amalgamations and HNN extensions over finite or virtually cyclic groups. However, we show that not all groups admitting such a hierarchy have logarithmic separation profile by providing an example of a surface amalgam over a cyclic group with superlogarithmic separation profile.

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 prove that hyperbolic groups with logarithmic separation profiles split over cyclic groups. result 0.94 0.95 0.93 asserted 1.00
s1 This shows that such groups can be inductively built from Fuchsian groups and free groups by amalgamations and HNN extensions over finite or virtually cyclic groups. interpretation 0.90 0.81 0.13 asserted 0.97
s2 However, we show that not all groups admitting such a hierarchy have logarithmic separation profile by providing an example of a surface amalgam over a cyclic group with superlogarithmic separation profile. result 0.80 0.95 0.08 asserted 1.00

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

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  • references and citing sentences
  • relations to other papers
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  • 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 prove that hyperbolic groups with logarithmic separation profiles split over cyclic groups. This shows that such groups can be inductively built from Fuchsian groups and free groups by amalgamations and HNN extensions over finite or virtually cyclic groups. However, we show that not all groups admitting such a hierarchy have logarithmic separation profile by providing an example of a surface amalgam over a cyclic group with superlogarithmic separation profile.

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