Welcome to MathOverflow

Question

0

Consider a finitely presented group $G$ with presentation $P$ given by $\left\langle g_1,\ldots,g_n|\, r_1,\ldots,r_m\right\rangle$, equipped with a homomorphism $\rho\colon\, G\to H$ to a finitely generated group $H$ (generators in the presentation are ordered). Perform a sequence of Tietze transformations starting at $P$ and ending at presentation $P^\prime$ given by $\left\langle g^\prime_1,\ldots,g^\prime_n|\, r^\prime_1,\ldots,r^\prime_m\right\rangle$ which differs from $P$ by relabeling of generators (so $g_i\mapsto g_i^\prime$ for all $i$ is an isomorphism of presentations). Is it true that $\rho(g_i)=\rho(g_i^\prime)$ for all $i$ up to inner automorphism of $H$?

At first I thought this was trivial, but I'm worried that it might actually be false- if so, is there some way to measure such a failure of simple-connectedness of "presentations of $G$ over $H$".

edited Jul 22 at 14:43

asked Jul 19 at 7:19

Daniel Moskovich
11.1k●1●36●106

Daniel: (i) Have you an example of such a sequence of Tietze transformation that relabel generators only? (ii) Have you thought of using Fox derivatives or something of that form? – Tim Porter Jul 19 at 8:19

2

So, let $G$ be a free group $\langle g_1,g_2\rangle$ and consider the alternative presentation $\langle g_2,g_1\rangle$. Now consider the obvious map $\rho:G\to \mathbb{Z}/2\times\mathbb{Z}/2$. – HW Jul 19 at 8:25

2

In view of HW's example, you have to allow all automorphisms of $H$, not just the inner ones. However, the statement is false even in this form: Take $G=F_n$. You would be asking if the action of $Out(F_n)$ on $Hom(G,H)/Aut(H)$ is trivial. This is clearly false too, say, for $G=SL(2,p)$, $p\ge 5$. (Same argument as in the case of the $SL(2,C)$-character varieties.) – Misha Jul 19 at 12:29

... Oh dear. So it is false. Misha: What is the argument for $SL(2,C)$ character varieties? Does the statement have a chance of being true for fundamental groups of (cusped) hyperbolic 3-manifolds? – Daniel Moskovich Jul 19 at 13:32

@Dan: Think first about Dehn twists acting on the character variety (and why mapping class group acts nontrivially on the Teichmuller space). The easiest case to consider is when $F_3=F_2* Z$, then apply $\phi\in Aut(G)$, where $\phi|F_2=id$ and $\phi|Z$ is conjugation by $g_1\in F_2$. Same works if $G=F_2$, use $(g_1,g_2)\to (g_1, g_1g_2)$. – Misha Jul 20 at 5:51

show 2 more comments

Daniel Moskovich · Accepted Answer · 2012-07-22 14:42:30Z UTC

The question is effectively answered by HW and by Misha in the comments. Namely, a sequence of Tietze transformations starting and ending at "the same" presentation defines an automorphism of $G$. But the action of $\mathrm{Out}(G)$ on $\mathrm{Hom}(G,H)/\mathrm{Aut}(H)$ is non-trivial in general.

Monodromy in presentations of one group over another

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.