Examples of group families with solvable uniform word problem I would like to know of any examples of families of groups that are known (or conjectured) to have a solvable uniform word problem, i.e. an algorithm that given a presentation $P$ of a group in the family, and a word $W$ in the generators of $P$, decides whether $W$ represents the identity of the group defined by $P$.
In particular, I'm curious about finitely generated Fuchsian groups.
 A: Derek Holt is the expert here - I hope he will correct me where I err.

There are many such families.  Here are a few ones that spring to mind (or were mentioned in the comments above), in no particular order.

*

*fuchsian groups

*fundamental groups of three-manifolds

*hyperbolic groups

*automatic groups

*one-relator groups

*residually finite groups


Here is the general idea for families (1) to (5).  Suppose that $\mathcal{F}$ is a family of (finitely presentable) groups.  Suppose also that $\mathcal{F}$ comes with a solution to the "certification problem".  That is, there is an algorithm that, given a finite presentation $P$ of a group $G \in \mathcal{F}$, produces a (useful!) proof that $G$ lies in $\mathcal{F}$.  We then use this proof to solve the word problem for $P$.

Here is an example in more detail.  Suppose that $\mathcal{F_3}$ is the family of fundamental groups of closed, connected three-manifolds.  Given a presentation $P$ (promised to be of a group in $\mathcal{F_3}$), our algorithm returns

*

*a one-vertex triangulation $T$ of a three-manifold $M$ and

*a sequence of Tietze moves connecting $P$ to the triangulated presentation of $\pi_1(M)$ coming from $T$.

The existence of an algorithm for the word problem for $\pi_1(M)$ (and thus $P$) follows from geometrisation.

Another example may be integer linear groups, but there is a subtle point.  We need an algorithm that, given a finite collection of invertible integer matrices, produces a finite presentation of the group they generate.  Does such an algorithm exist?
EDIT: As pointed out by HJRW (in the comments) there is no such algorithm.  This is proven by Bridson and Wilton.
