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Freely indecomposable groups acting on hyperbolic spaces

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Ilya Kapovich, Richard Weidmann

Group Theory · Submitted 2 Mar 2002

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

The authors' main claim We obtain a number of finiteness results for groups acting on Gromov-hyperbolic spaces.

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arXiv IDmath/0203015 (version 2)
Pith IDW44C424C, Pith's permanent record of this paper and its review
CitationIlya Kapovich, Richard Weidmann. "Freely indecomposable groups acting on hyperbolic spaces." arXiv:math/0203015 (2002). https://pith.science/paper/W44C424C
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,060 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.36 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.31 · compares with prior results 0.86 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We obtain a number of finiteness results for groups acting on Gromov-hyperbolic spaces. In particular we show that a torsion-free locally quasiconvex hyperbolic group has only finitely many conjugacy classes of $n$-generated one-ended subgroups. We also show that the rank problem is solvable for the class of torsion-free locally quasiconvex hyperbolic groups (even though it is unsolvable for the class of all torsion-free hyperbolic groups). We apply our results to 3-manifold groups. Namely, suppose $G$ is the fundamental group of a closed hyperbolic 3-manifold fibering over a circle and suppose that all finitely generated subgroups of $G$ are topologically tame. We prove that for any $k\ge 2$ the group $G$ has only finitely many conjugacy classes of non-elementary freely indecomposable $k$-generated subgroups of infinite index in $G$.

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 obtain a number of finiteness results for groups acting on Gromov-hyperbolic spaces. contribution 0.61 0.96 0.27 asserted 1.00
s1 In particular we show that a torsion-free locally quasiconvex hyperbolic group has only finitely many conjugacy classes of $n$-generated one-ended subgroups. result 1.00 0.96 0.36 asserted 1.00
s2 We also show that the rank problem is solvable for the class of torsion-free locally quasiconvex hyperbolic groups (even though it is unsolvable for the class of all torsion-free hyperbolic groups). result 0.97 0.92 0.18 asserted 1.00
s3 We apply our results to 3-manifold groups. contribution 0.88 0.89 0.03 asserted 0.94
s4 Namely, suppose $G$ is the fundamental group of a closed hyperbolic 3-manifold fibering over a circle and suppose that all finitely generated subgroups of $G$ are topologically tame. background 0.51 0.11 0.03 unclear 0.42
s5 We prove that for any $k\ge 2$ the group $G$ has only finitely many conjugacy classes of non-elementary freely indecomposable $k$-generated subgroups of infinite index in $G$. result 0.99 0.95 0.30 asserted 1.00

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

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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 obtain a number of finiteness results for groups acting on Gromov-hyperbolic spaces. In particular we show that a torsion-free locally quasiconvex hyperbolic group has only finitely many conjugacy classes of $n$-generated one-ended subgroups. We also show that the rank problem is solvable for the class of torsion-free locally quasiconvex hyperbolic groups (even though it is unsolvable for the class of all torsion-free hyperbolic groups). We apply our results to 3-manifold groups. Namely, suppose $G$ is the fundamental group of a closed hyperbolic 3-manifold fibering over a circle and suppose that all finitely generated subgroups of $G$ are topologically tame. We prove that for any $k\ge 2$ the group $G$ has only finitely many conjugacy classes of non-elementary freely indecomposable $k$-generated subgroups of infinite index in $G$.

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