I'm not sure if this question is more suited to mathoverflow.

Is there a systematic way to solve for the automorphism group of a finitely presented group with a solvable word problem? Preferably with software.

Here is my particular example:

$$G_n = \langle r_1, r_2, \dots , r_n \; \vert \;r_i^{n_i}, r_1 r_2 \dots r_n \rangle \quad \text{for some} \quad \{n_i\} \subset\mathbb N$$

This is a type of hyperbolic reflection group generated by a finite number of rotations. They are index $2$ subgroups of coxeter groups.

I've tried using the GAP System for Computational Discrete Algebra to find the automorphism group, but the method AutomorphismGroup doesn't seem to terminate. However, a rewriting system for all such groups can be developed using the kbmag package.

I'm actually interested in the outer automorphisms, as these are nontrivial. I expect the rank of $\mathrm{Out}(G_4)$ to be $2$, and I think I have the explicit form of the generators in this case.

However, I don't know how to check whether my suspected generators do in fact generate the whole outer automorphism group, and so I seek a way to compute $\mathrm{Out}(G_i)$ for given rotation orders $n_i$.

  • $\begingroup$ This is not a solvable group in general. $\endgroup$ – Lee Mosher Mar 17 '17 at 15:25
  • $\begingroup$ @LeeMosher - Unless I've made a typo, they certainly are. These are index 2 subgroups of coxeter groups which are solvable. $\endgroup$ – Myridium Mar 17 '17 at 15:27
  • $\begingroup$ @LeeMosher - Sorry, I confused the terminology of solvable group with solvable word problem! I will correct it. $\endgroup$ – Myridium Mar 17 '17 at 15:46
  • $\begingroup$ Ah, okay, I see. $\endgroup$ – Lee Mosher Mar 17 '17 at 15:52

I doubt there exist such general algorithms as you ask for.

However, I think this particular outer automorphism group can be identified using some topology. It is very close related to the "pure" mapping class group of the $n$-times punctured sphere $X = S^2 - \{p_1,...,p_n\}$, meaning the group of homeomorphisms preserving each puncture, modulo isotopy. Your group is isomorphic to the fundamental group of the compact 2-orbifold having underlying space $S^2$ and with cone points at $p_1,...,p_n$ having angles $2 \pi / n_1$,...,$2 \pi / n_n$ (subject to disambiguating $n$). In particular every homeomorphism of $X$ that preserves each puncture determines an outer automorphism of your group, and from this you get an injective homomorphism from the pure mapping class group of $X$ to the outer automorphism group of your group. I'm also pretty sure that this injective homomorphism has finite index image (and is in fact an isomorphism if the exponents $n_1,...,n_n$ are pairwise distinct; one needs to prove that the generating set $r_1,...,r_n$ is permuted up to conjugacy, and this is the one place where I am unsure...).

With these ideas, it follows that there are a lot of interesting generators which can be visualized using topology, and then written down very concretely. For instance, there is a circle on the sphere separating the points $p_1,...,p_k$ from the points $p_{k+1},...,p_n$ and representing the group element $t = r_1 \cdots r_k$. Doing a Dehn twist around that circle represents the outer automorphism class of the automorphism fixing $r_1,...,r_k$ and mapping $r_i \to t r_i t^{-1}$ for $i=k+1,...,n$.

I'll say that these ideas are all closely related to the circle of ideas surrounding the Dehn-Nielsen-Baer theorem which says that if $S$ is a compact, connected, oriented surface of genus $\ge 1$ then $\text{Out}(\pi_1 S)$ is isomorphic to the mapping class group of $S$. I believe that much of the Dehn-Nielsen-Baer theorem extends to compact, hyperbolic 2-orbifolds, although finding a reference to this might be difficult.

Added: You also asked about half-twist generators. Again, these can be described pretty much straight from the topological picture. For instance, the group element $t = r_1 r_2$ represents a circle separating $p_1,p_2$ from the rest of the cone points. Assuming the orders of $p_1,p_2$ are equal, a half-twist around that circle is represented by the outer automorphism class of the automorphism $r_1 \to r_1 r_2 \bar r_1$, and $r_2 \to r_1$, and $r_i \to r_i$ for $i=3,...,n$.

  • $\begingroup$ I've looked into Mapping Class Groups and my research is in hyperbolic orbifolds. I believe you are right that the Mapping Class Group is in one-to-one correspondence with the things in $Out(G)$ when the cone points are distinct. I have a good resource (Mapping Class Groups by Farb and Margalit) to deal with this situation. My interest is in the less trivial cases where we can perform half Dehn twists; i.e. swapping cone points which have the same order. I can make educated guesses about the likely generators, but seek to confirm my guesses and know the exact group structure. $\endgroup$ – Myridium Mar 17 '17 at 16:18
  • $\begingroup$ @Myridium: DNB theorem holds for all closed 2d orbifolds except in the case of finite cyclic fundamental group. In the most interesting, hyperbolic case, this is an easy application of the Douady-Earle extension theorem. $\endgroup$ – Moishe Kohan Mar 17 '17 at 16:26
  • $\begingroup$ @MoisheCohen - The problem is that I do not know $\mathrm{Homeo}(S)$ when $S$ is an orbifold. Only homeomorphisms which swap cones of equal order are allowed. All other cones/punctures must be fixed. $\endgroup$ – Myridium Mar 17 '17 at 16:40
  • $\begingroup$ @MoisheCohen: Thanks for that reference to Douady-Early. $\endgroup$ – Lee Mosher Mar 17 '17 at 17:01
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    $\begingroup$ @Myridium: Yes, of course. You can think of this group as a certain finite index subgroup of the mapping class group of the punctured surface. $\endgroup$ – Moishe Kohan Mar 17 '17 at 17:02

There are a number of algorithms for automorphism group computations of finite groups. I am not aware of a generic algorithm for infinite groups.

I suspect that for infinite polycyclic groups it might be possible (modulo some potentially nasty number theory) to generalize the methods for finite solvable groups, but that is a research-level problem without a turn-key implementation.

When you call AutomorphismGroup in GAP on a finitely presented group, it will try to test finiteness first, this is probably the non-terminating calculation you observe.


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