Questions in Geometric Group Theory — Q 1.12

Definitions and mathematical statement

A finitely generated group $\Gamma$ is hyperbolic if, for a finite generating set, its Cayley graph with the word metric has $\delta$-thin geodesic triangles for some $\delta\geq0$. It is one-ended if the complement of every finite vertex set in this graph has exactly one infinite connected component. For $L\leq\Gamma$, the index $[\Gamma:L]$ is the number of left cosets of $L$.

Question (Whyte). Does there exist a one-ended hyperbolic group $\Gamma$ and subgroups $H,K\leq\Gamma$ such that \[ [\Gamma:H]<\infty,\qquad [\Gamma:K]=\infty,\qquad H\cong K? \] Thus the question concerns the possibility of such a group, not an assertion that every one-ended hyperbolic group has such subgroups. The supplied research summary reports an affirmative answer in the literature and attributes an example to Stark--Woodhouse; this is the record's reported status, not a proof supplied here.

Short English statement

Can one one-ended hyperbolic group contain isomorphic subgroups of finite and infinite index? The record reports that such examples exist.

Sources

• M. Bestvina, Questions in Geometric Group Theory (2004), Question 1.12, PDF page 3, attributed there to Whyte: https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf. The record describes manual transcription and access on 2026-07-29.
• The supplied research summary attributes an affirmative example to Stark--Woodhouse, section 2, without a title, year, or bibliographic identifier. Its sketch also cites Bestvina--Feighn for hyperbolicity and Neumann's covering lemma; no further references are supplied.
• Local registry data/unsolvedmath/problems.json, permanent ID $1100112$, source code catalogue entry, Questions in Geometric Group Theory --- Q 1.12. The record labels this solved and SOLVED-IN-LITERATURE; the source-list status and rights checks remain marked NEEDS_REVIEW.