research/AMR-010-0119.md

metadata

id: AMR-010-0119
classification: SOLVED-IN-LITERATURE
wording_corrected: 'no'

AMR-010-0119 — Mitra's question: does inclusion of hyperbolic groups extend to a Cannon–Thurston map of boundaries?

Problem (corrected statement if needed)

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):

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 $\partial H \to \partial G$?

Such an extension, when it exists, is called a Cannon–Thurston (CT) map. No correction of the wording is needed.

Status / Literature

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.

Positive results (map exists):

• 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).

Negative result (the resolution):

• 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).

Work done

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.

Result

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.

What remains

• 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.