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| id: AMR-010-0119 |
| classification: SOLVED-IN-LITERATURE |
| wording_corrected: no |
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| # AMR-010-0119 — Mitra's question: does inclusion of hyperbolic groups extend to a Cannon–Thurston map of boundaries? |
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| ## Problem (corrected statement if needed) |
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| The worklist transcription matches the source verbatim (verified against the original PDF at |
| https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf, Question 1.19, "Maps Between Boundaries", PDF page 5): |
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| > **Q 1.19 (M. Mitra).** Let $G$ be a word-hyperbolic group and $H$ a word-hyperbolic subgroup. Does the inclusion $H \to G$ extend to a continuous map between the boundaries $\partial H \to \partial G$? |
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| Such an extension, when it exists, is called a **Cannon–Thurston (CT) map**. No correction of the wording is needed. |
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| ## Status / Literature |
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| **Answer: NO in general.** The question was answered negatively by Baker and Riley in 2013. All citations below were verified against Crossref, the arXiv API, or publisher pages during this work. |
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| Positive results (map exists): |
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| - **Quasi-convex (undistorted) $H$:** a quasi-isometric embedding of hyperbolic spaces extends to a topological embedding of Gromov boundaries — classical (Gromov; see e.g. Kapovich–Benakli, *Boundaries of hyperbolic groups*, Contemp. Math. 296 (2002), DOI 10.1090/conm/296/05068, verified via Crossref reference data). |
| - **Cannon–Thurston (1985/2007):** the original example — the fiber surface group of a closed hyperbolic 3-manifold fibering over $S^1$; the CT map is a group-equivariant Peano curve $S^1 \twoheadrightarrow S^2$. J. W. Cannon and W. P. Thurston, *Group invariant Peano curves*, Geom. Topol. 11 (2007), 1315–1355, DOI 10.2140/gt.2007.11.1315 (verified via Crossref reference data in 10.1017/fms.2013.4). |
| - **Mitra 1998a:** CT maps exist when $H$ is an infinite normal subgroup of a hyperbolic $G$ (in particular for hyperbolic group extensions). M. Mitra, *Cannon–Thurston maps for hyperbolic group extensions*, Topology 37(3) (1998), 527–538, DOI 10.1016/S0040-9383(97)00036-0 (**verified via Crossref**). Point preimages are described by an ending-lamination theory: M. Mitra, *Ending laminations for hyperbolic group extensions*, GAFA 7(2) (1997), 379–402, DOI 10.1007/PL00001624 (verified via Crossref reference data). |
| - **Mitra 1998b:** CT maps exist for vertex/edge groups in trees of hyperbolic spaces with quasi-isometric edge-to-vertex monomorphisms — this is the [Mit98b] cited in Bestvina's Q 1.19. M. Mitra, *Cannon–Thurston maps for trees of hyperbolic metric spaces*, J. Differential Geom. 48(1) (1998), 135–164, DOI 10.4310/jdg/1214460609 (**verified via Crossref**). |
| - **Mj 2014:** CT maps exist for simply/doubly degenerate surface Kleinian groups (settling a question of Cannon–Thurston and Thurston's 1982 Problem 14); later for arbitrary finitely generated Kleinian groups (McMullen's conjecture; arXiv:1002.0996). M. Mj, *Cannon–Thurston maps for surface groups*, Ann. of Math. (2) 179(1) (2014), 1–80, DOI 10.4007/annals.2014.179.1.1 (**verified via the Annals of Mathematics journal page**). |
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| Negative result (the resolution): |
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| - **Baker–Riley 2013:** O. Baker and T. R. Riley, *Cannon–Thurston maps do not always exist*, Forum Math. Sigma 1 (2013), Paper No. e3, 11 pp., DOI 10.1017/fms.2013.4 (**verified via Crossref**; abstract: "We construct a hyperbolic group with a hyperbolic subgroup for which inclusion does not induce a continuous map of the boundaries"), arXiv:1206.0505 (**verified via arXiv API**), MR3143716. This settles Q 1.19 in the negative. |
| - **Matsuda–Oguni:** building on Baker–Riley, every non-elementary hyperbolic group embeds in *some* hyperbolic group with no CT map. Y. Matsuda and S. Oguni, *On Cannon–Thurston maps for relatively hyperbolic groups*, arXiv:1206.5868 (cited in the Baker–Riley paper; verified only as a cited arXiv preprint, not independently fetched). |
| - **Distortion dichotomy:** subexponentially distorted subgroups of hyperbolic groups are quasi-convex (I. Kapovich, *The combination theorem and quasiconvexity*, Internat. J. Algebra Comput. 11(2) (2001), DOI 10.1142/S0218196701000553, verified via Crossref reference data), so CT maps exist there; Baker–Riley also showed CT maps can exist for extremely (non-recursively) distorted free subgroups of hyperbolic hydra: O. Baker and T. Riley, *Cannon–Thurston maps, subgroup distortion, and hyperbolic hydra*, Groups Geom. Dyn. 14 (2020) (bibliographic details seen in reference lists of later papers; arXiv:1209.0815 cited within the Baker–Riley paper itself; DOI not independently fetched). |
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| ## Work done |
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| 1. **Source identification and wording check.** Fetched the Bestvina "Questions in Geometric Group Theory" PDF (July 2004 update) and confirmed Q 1.19 appears exactly as transcribed, on PDF page 5 in §1.6 "Maps Between Boundaries". The worklist text is a faithful transcription; `wording_corrected: no`. |
| 2. **Citation verification.** Verified the Baker–Riley paper directly via Crossref (10.1017/fms.2013.4) and the arXiv API (arXiv:1206.0505 — note: my first two guesses at the arXiv identifier, 1206.1482 and 1206.5368, returned unrelated physics/CS papers, a useful reminder that unverified identifiers are worthless); verified both Mitra 1998 papers via Crossref; verified Mj's Annals paper via the journal page. Cannon–Thurston 2007, Mitra 1997 (GAFA), and Kapovich 2001 were verified as DOI-asserted references inside the Crossref records fetched. |
| 3. **Extracted and checked the Baker–Riley argument** from the full text (arXiv:1206.0505v4). The construction: |
| - Let $C, C_i$ (on $c_1, c_2$) and $D_j, D_{ij}$ (on $d_1, d_2$) be long Rips-type positive words (e.g. $C = c_1 c_2 c_1 c_2^2 c_1 c_2^3 \cdots c_1 c_2^r$). For $r$ large, |
| $$G = \langle a, b, c_1, c_2, d_1, d_2 \mid a^{-1}b^{-1}ab = C,\ b^{-1}c_i b = C_i,\ (ab)^{-1} d_j (ab) = D_j,\ c_i^{-1} d_j c_i = D_{ij} \rangle$$ |
| satisfies $C'(1/6)$, hence is hyperbolic (in fact a CAT($-1$) variant exists — Remark 9 of the paper, using Wise's pentagon pieces). |
| - $H = \langle b, d_1, d_2 \rangle$ is free of rank 3: the presentation is a tower of HNN extensions ($F(d_1,d_2) \leadsto G_{cd} \leadsto G_{bcd} \leadsto G$) and Britton's lemma, together with $F(c_1,c_2) \cap F(d_1,d_2) = \{1\}$ in $G_{cd}$, rules out any relation among $b, d_1, d_2$. |
| - **No CT map.** Mitra's criterion (Lemma 6 of the paper): the CT map exists iff $M(N) \to \infty$, where $M(N)$ measures how far $G$-geodesics between endpoints of $H$-geodesics staying outside $B(N)$ in $X_H$ must stay from $e$ in $X_G$. The words $w_n = b^{-n} a^{-n} d_1 a^{n} b^{n}$ are strongly Dehn-reduced, hence (Lemma 7, Kapovich–Short, via Strebel's appendix to Ghys–de la Harpe) label *geodesics* in the Cayley graph of $G$ passing through $e$ — so their endpoint pair is at $G$-distance $0$ from the identity along the path. But the relations rewrite $a^n b^n$ as a positive word in $ab, c_1, c_2$, and thence $w_n = u^{-1} d_1 u$ with $u^{-1}d_1u$ a positive word in $d_1, d_2$: the endpoints lie in $H$, and the $H$-geodesic between them stays at distance $\geq n$ from $e$ in $X_H$. Hence $M(n) = 0$ for all $n$, $M(N) \not\to \infty$, and no continuous extension $\partial H \to \partial G$ exists. |
| 4. **Why the positive theorems do not save the question.** The example clarifies the boundary of Mitra's theorems: $H$ is *not* normal in $G$, and in the factorization $H \hookrightarrow G_{bcd} \hookrightarrow G$ the middle group $G_{bcd}$ is hyperbolic and an HNN extension, but the defining monomorphisms fail the quasi-isometric-embedding hypothesis of the trees-of-spaces theorem — Baker–Riley show (Remark 8) that in fact *both* inclusions $H \hookrightarrow G_{bcd}$ and $G_{bcd} \hookrightarrow G$ admit no CT map, so the q.i.-embedding hypothesis in Mitra's tree theorem is essential, not an artifact. Also, $H$ has infinite height in $G$, so the example is consistent with Swarup's finite-height quasiconvexity question (Q 1.8 on Bestvina's list) remaining open. |
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| ## Result |
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| Mitra's question (Bestvina Q 1.19) is **resolved in the negative**: Baker and Riley (Forum Math. Sigma 1, 2013, e3; DOI 10.1017/fms.2013.4; arXiv:1206.0505) constructed an explicit $C'(1/6)$ small-cancellation hyperbolic group $G$ on six generators containing a rank-3 free subgroup $H = \langle b, d_1, d_2 \rangle$ for which no Cannon–Thurston map $\partial H \to \partial G$ exists, with an elementary, fully rigorous proof via Mitra's $M(N)$ criterion and Dehn-reduced geodesics. Hence the answer to the question as posed is **no**, and the problem is solved in the literature. This is a literature triage, not an independent solution by me; I verified the source wording, the resolving paper's existence and abstract against Crossref and the arXiv API, and reconstructed its proof from the full text. |
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| ## What remains |
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| - **Characterization problem:** given hyperbolic $H \leq G$, decide when a CT map exists. Known sufficient conditions: quasi-convexity; normality (Mitra); tree-of-spaces with q.i. edge maps (Mitra); Kleinian groups (Mj). Baker–Riley shows distortion alone is not the criterion: their $H$ is at least doubly-exponentially distorted, while hyperbolic hydra contain even more distorted free subgroups *with* CT maps. Subexponential distortion forces quasi-convexity (Kapovich), so the remaining gap is **Kapovich's question: is there an exponentially distorted hyperbolic subgroup of a hyperbolic group with no CT map?** (explicitly left open in Baker–Riley). |
| - **Structure of CT maps when they exist:** description of point preimages beyond the normal-extension/ending-lamination case (this is Bestvina Q 1.20, attributed to Swarup). |
| - **Relatively hyperbolic/generalizations:** CT maps for relatively hyperbolic groups and their subgroups (Matsuda–Oguni arXiv:1206.5868; later work of Mj–Pal and others), for CAT(0) groups with isolated flats, and non-existence results in the hierarchically hyperbolic setting are active topics (recent literature through 2025–2026 still cites Baker–Riley as the foundational counterexample). |
| - Related open items on the same Bestvina list touched by this example: Swarup's finite-height question (Q 1.8) and the point-preimage problem (Q 1.20) remain open as far as I could verify within the fetch budget. |
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