HYPERBOLIC AND AUTOMATIC GROUPS

See also problems (06), (OR10).

(H1) (a) Are  hyperbolic groups residually finite?

(b) Does every hyperbolic group have a proper subgroup of finite index ?   Background

*(H2) Are hyperbolic groups linear?   Background

(H3) Do hyperbolic groups with torsion have solvable isomorphism problem?   Background

(H4) (A.Miasnikov) Given a finite presentation of a hyperbolic group (which is not necessarily a Dehn presentation), is it possible to find a Dehn presentation for this group in polynomial time?

(H5) (A.Miasnikov) Given a finite presentation of an automatic group, can one decide if this group is hyperbolic?  Background

(H6) (S.Gersten) Are all automatic groups biautomatic? Background

(H7) (S.Gersten)  Does every automatic group have a solvable conjugacy problem?  Background

(H8) (S.Gersten)  Is every biautomatic group which does not contain any Z x Z subgroups, hyperbolic?  Background

(H9) (S.Gersten)  Can the group  <x,y;  yxy^{-1} = x^2>  be a subgroup of an automatic group?  Background

(H10) (S.Gersten)  Is a retract of an automatic group automatic?  Background

(H11) Does every hyperbolic group act properly discontinuously and co-compactly by isometries on a CAT(k) space, where  k<0?

(H12) (G.Baumslag, A.Miasnikov, V.Remeslennikov) Is every hyperbolic group equationally noetherian?  (A group is called equationally noetherian if every system of equations in this group is equivalent to a finite subsystem). Background

(H13) Are combable groups automatic?

*(H14) (A.Miasnikov) We call a subgroup  H  of a group  G  malnormal if for any element  g  in G, but not in H, one has  H^g \cap H  = {1}.  Is this property algorithmically decidable for finitely generated subgroups of hyperbolic groups?  Background

(H15) (A.Miasnikov) If a finitely generated subgroup  H  of a hyperbolic group is malnormal (see above), does it follow that  H  is quasiconvex?

(H16)  Is every metabelian automatic group virtually abelian ?

(H17) (A.Olshanskii) Does every hyperbolic group H have a free normal subgroup F such that the factor group H/F is of finite exponent?

(H18) (S.Gersten)  Let F be Thompson's group:
(x_0, x_1, ...   |   x_i x_k x_i^{-1} = x_{k+1},   k>i,   k=1,2,...).
Is F automatic? Background