Examples of locally hyperbolic groups It is well-known that a subgroup of a hyperbolic group need not be hyperbolic. Let us say that a (finitely generated) group $G$ is locally hyperbolic if all its finitely generated subgroups are (Gromov) hyperbolic. I can think of two families of examples of locally hyperbolic groups:

*

*Free groups; this follows from the Nielsen-Schreier theorem.

*Surface groups of genus $g > 1$; this follows from the fact that any subgroup of such a surface group is either a surface group (automatically also of genus $g>1$) or else a free group.

Are there other natural classes? Are there any known properties for locally hyperbolic groups (e.g. decidable subgroup membership problem)?
 A: Many examples can be exhibited using a theorem of Gersten:
Theorem (Gersten): Let $G$ be a hyperbolic group of cohomological dimension 2. Every finitely presented subgroup $H$ of $G$ is hyperbolic.
This is already very interesting, but the question asked for examples where every finitely generated subgroup is word-hyperbolic. Recall that $G$ is called coherent if every finitely generated subgroup is finitely presented.
Corollary: If $G$ is word-hyperbolic, coherent, and of cohomological dimension 2, then every finitely generated subgroup of $G$ is hyperbolic.
We can now search the literature for examples of such groups:


*If $\Gamma$ is the mapping torus of a free-group endomorphism then it is 2-dimensional, and coherent by a theorem of Feighn--Handel. It is hyperbolic whenever the endomorphism is atoroidal, by work of Brinkmann (for automorphisms) and Mutanguha (in the general case). (Giles also mentions these in his answer.)


*A one-relator group $\Gamma=F/\langle\langle w\rangle\rangle$ has (virtual) cohomological dimension at most 2 by Lyndon and Cockroft, so in the context of your question we need to understand which are hyperbolic and which are coherent. Baumslag conjectured that such $\Gamma$ is always coherent, while Gersten conectured that $\Gamma$ is hyperbolic unless it contains a Baumslag--Solitar subgroup. These conjectures have been confirmed in certain cases:



*

*If $\Gamma$ has torsion (i.e. $w$ is a proper power) then $\Gamma$
is hyperbolic by the B.B. Newman Spelling theorem, and is coherent
by recent work of Wise and Louder--W.


*If $\Gamma$ is 2-free, i.e. every 2-generated subgroup of $\Gamma$ is free, then $\Gamma$ is hyperbolic by very recent work of Linton, and coherent by another recent paper of Louder--W.


*It is well known that random groups

$\Gamma = \langle a_1,\ldots, a_m\mid w_1,\ldots,w_n\rangle$
are a.a.s. hyperbolic in many regimes. They are also coherent by a recent paper of Kielak--Kropholler--Wilkes whenver $n<m-1$, and coherent with positive probability when $n=m-1$. It is reasonable to conjecture that they are still a.a.s. coherent when $n=m-1$, but a.a.s incoherent when $n\geq m$, and also in the density model.
As far as I know, the membership problem remains open in all these cases. It certainly isn't known for general locally hyperbolic groups.
A: 

*Finite groups.


*...


*...


*Fundamental groups of closed connected hyperbolic three-manifolds.
Here is a proof of the latter using far too many tools from kleinian groups.  Fundamental groups of closed connected hyperbolic three-manifolds are quasi-isometric to hyperbolic three-space and thus are Gromov hyperbolic. This deals with all finite index subgroups of $\pi_1(M)$.
Suppose instead that $N$ is an infinite cover of $M$, with $\pi_1(N)$ finitely generated.  Applying the tameness theorem (Agol, and also Calegari-Gabai) and the covering theorem (Canary) we have that $N$ is either completely degenerate (and homotopic to a closed surface) or has flaring ends.  In the latter case $\pi_1(N)$ is quasi-convex in $\pi_1(M)$ and thus is Gromov hyperbolic.
A: A locally hyperbolic group is in particular coherent (i.e. locally finitely presented), which is already a special property. To add to the examples already given by Sam Nead: ascending HNN extensions of free groups (a.k.a. mapping tori of free group endomorphisms), such as free-by-cyclic groups, were shown to be coherent by Feighn and Handel. Recently, Mutanguha has shown that indeed if such a group is hyperbolic, then it is locally hyperbolic (see Corollary 7.6 in his paper).
Feighn, Mark; Handel, Michael, Mapping tori of free group automorphisms are coherent, Ann. Math. (2) 149, No. 3, 1061-1077 (1999). ZBL0938.20022.
Mutanguha, Jean Pierre, The dynamics and geometry of free group endomorphisms, Adv. Math. 384 (2021),  ZBL07415144 arXiv:2005.11896.
