Every finitely generated group is a colimit of a sequence of finitely presented groups. If every group in the sequence is residually finite what can one say about the colimit?


More precisely:
Is the colimit of a sequence of residually finite groups again residually finite?



Colimits of residually finite groups
(a) There exists a finitely generated (f.g.) group that is an inductive limit of a sequence of epimorphisms of f.g. residually finite groups, and is not residually finite. The first such example is due to B.H. Neumann (1937) and I describe it below.


Write $Y=\{(i,j)\in\mathbf{Z}\times\mathbf{N}: |i|\le j\}$. View it as the disjoint union of the finite subsets $Y\_j=\{(i,j): |i|\le j\}$; write $Y\_{\ge m}=\bigcup\_{j\le m}Y\_j$. Let $W\_m$ be the group of permutations of $Y\_{\ge m}$ preserving each $Y\_j$: it is residually finite (since it embeds in the (infinite) product of symmetric groups on the $Y\_j$ for $j\ge m$ (note that $W\_m$ is quotient of $W\_{m-1}$ by a finite normal subgroup).


Let $t$ act on each $Y\_j$, $j\ge 1$ as the transposition $0\leftrightarrow 1$. Let $c$ act on each $Y\_j$ as the cycle $(-j,\dots,j)$. On $Y\_j$, the images of $c$ and $t$ generate the symmetric group (on $2j+1$ letters). Then $c$ and $t$ define elements $c\_m$, $t\_m$ of $W\_m$, acting on $Y\_{\ge m}$. Define $G\_m=\langle c\_m,t\_m\rangle\subset W\_m$. Then clearly the image of $G\_{m-1}$ in $W\_m$ by the canonical projection ("forgetting $Y\_{m-1}$") is $G\_m$.


So we have epimorphisms $G\_2\to G\_3\to\dots$, and all are residually finite. Let $G$ be the inductive limit. The kernel of $G\_2\to G\_m$ is the set of elements of $G\_2$ that are identity outside $\bigcup\_{2\le j\le m}Y\_j$. Hence the kernel of $G\_2\to G$ the union of these kernels, namely the set of elements of $G\_2$ with finite support (typical nontrivial such elements are $[t,c^ntc^{-n}]$ for $n\ge 2$). It is not hard to see that we can identify the quotient $G$ as the group acting on $\mathbf{Z}$ generated by the transposition $0\leftrightarrow 1$ and the shift $n\mapsto n+1$. This 2-generated group contains the group of finitely supported permutations of $\mathbf{Z}$, which is not residually finite, since it has an infinite simple subgroup of index 2 (Onofri 1929).


This example was rediscovered when the concept of LEF group was clarified (Stëpin in the early 1980s, Vershik-Gordon in the 1990s).




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 (b) Edit 2: Here's an additional example with an inductive limit of finitely presented groups. [Edit 3: the example in (a) might work similarly, see (c) below]
For $n\le\infty$, let $M\_n$ be the infinitely generated "right-angled Artin group" $$\langle x\_m,y\_m:m\in\mathbf{Z}\mid [x\_p,x\_q],[x\_p,y\_q]\text{ if } 0<|p-q|<n\rangle.$$
Note that $M\_\infty$ is the restricted direct product $\bigoplus\_{p\in\mathbf{Z}}F(x\_p,y\_p)$ of the free groups on 2 generators .


Using the obvious shift automorphism $x\_p\mapsto x\_{p+1}$, $y\_p\mapsto y\_{p+1}$, define, for $n\le\infty$ the $H\_n$ as the semidirect product $M\_n\rtimes\mathbf{Z}$.


Then $H\_n$ has the presentation
$$\langle t,x\_0,y\_0\mid [x\_0,t^px\_0t^{-p}],[x\_0,t^py\_0t^{-p}]:1<|p|\le n\rangle. $$


This is a finite presentation for $n<\infty$. It is not hard to check in addition that this is a residually finite group.


But $H\_\infty$ is a standard wreath product $F(x\_0,y\_0)\wr\mathbf{Z}$, which is not residually finite by an old remark of Gruenberg (1957).




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Let's go back to (a), it might works too (i.e. with f.p. residually finite approximating groups). This is mostly similar to (b).


For $n\le\infty$, define the Coxeter group
$$Q\_n=\langle x\_p:p\in\mathbf{Z}\mid x\_p^2,(x\_px\_{p+1})^3\;\forall p,[x\_p,x\_q],\;\forall p,q\text{ s.t. }2\le |p-q|\le n\rangle.$$


Note that $Q\_\infty$ can be represented as the group of finitely supported permutations of $\mathbf{Z}$, $x\_i$ acting as transposition $i\leftrightarrow i+1$. Consider the automorphism $x\_i\mapsto x\_{i+1}$ of $Q\_n$ and define $L\_n=Q\_n\rtimes\mathbf{Z}$. Thus $L\_\infty$ is the B.H. Neumann group of (a), which can be also represented as the group of those permutations of $\mathbf{Z}$ that act as a translation outside a finite subset, generated by transposition $0\leftrightarrow 1$ and by the shift $p\mapsto p+1$.


The group $L\_n$ is presented, namely as
$$L\_n=\langle t,x\_0\mid x\_0^2,(x\_0tx\_0t^{-1})^3,[x\_0,t^px\_0t^{-p}]:2\le p\le n\rangle,$$
which for $n<\infty$ is a finite presentation. I believe it can be checked that $L\_n$ is residually finite if $n<\infty$. [The easy virtual retraction argument I have in mind in (b), however, doesn't work: Coxeter groups with odd exponents are a little more complicated that those with only even/infinity exponents.]


