4
votes
0answers
86 views

When does finite presentability of the associated graded Lie algebra of a group imply the group is finitely presented?

Let $G$ be a finitely generated group; let $L(G)$ denote the graded Lie algebra (over $\mathbb{Q}$) associated to the lower central series of $G$. I would like to know conditions for when the finite ...
7
votes
2answers
182 views

Modifying Dehn's algorithm to allow equal length replacements?

I'm an analyst trying to understand a certain class of finitely presented groups (one example is below) so it's quite likely this question is naive but I hope it is at least intelligible. Given a ...
7
votes
1answer
176 views

Minimal normally generating subsets of minimal generating sets

Let $G$ be a finitely generated group. The weight $w(G)$ of $G$ is defined to be the minimum number of elements of $G$ whose normal closure in $G$ is the whole of $G$ (this is sometimes also called ...
3
votes
1answer
94 views

Long words represent by multiplication of short words

Give a free group $G$ and one of its subgroup $H$ satisfies $rank(G)=n$$[G:H]=k$ Fix a generators of $G$ so we can talk about the length of elements in $G$.Then do there exist constants $A,B,C$ which ...
-1
votes
1answer
133 views

Homeomorphism of the punctured sphere which fixes an essential Jordan curve

$\phi$ is a homeomorphism from the 2-sphere to itself which represents an element of $PMCG(S^2,A)$ (we also denote it by $\phi$), where $A$ is a finite set of $S^2$. $\gamma$ is an essential Jordan ...
7
votes
2answers
347 views

Number of subgroups of a given index of a free group

Given $n,d\in \mathbb{Z}^+$, how many subgroups of index $d$ does the free group of rank $n$ have? In case $n=1$ the question is trivial, and in case $n=2, d=2$ there are 3 such subgroups. I think I ...
8
votes
0answers
184 views

Computing van Kampen diagrams

If G is a finitely presented group (with generating set X) and w is a word over X such that w=1 in G, then the latter can be witnessed by a so called van Kampen diagram for w, which is a planar ...
1
vote
1answer
180 views

how to classify epimorphisms from a subgroup to itself?

Assume $G$,$\hat{G}$ are both free group of rank $n$,and $H$,$\hat{H}$ be their subgroups of index $k$ respectively,$h:H \rightarrow G$, $\hat{h}:\hat{H} \rightarrow\hat{G}$, are two epimorphisms. We ...
8
votes
2answers
318 views

generators of free group

Give a rank $n$ free group $G=\langle a_1,a_2,\dots,a_n\rangle$, let $g_1,g_2,\dots,g_n \in G$, $b_j=g_j^{-1}a_jg_j$ . If $b_1,b_2,\dots,b_n$ can generates the whole $G$, what can we say about ...
5
votes
1answer
219 views

“Concretely” writing down elements in a free profinite group

Let $r$ be a natural number. The elements of the free group $F_r$ on $r$ generators have a nice concrete description as "words" in the $r$ generators (and their inverses). I'd like to know if there is ...
9
votes
1answer
373 views

Dehn's algorithm for word problem for surface groups

For some $g \geq 2$, let $\Gamma_g$ be the fundamental group of a closed genus $g$ surface and let $S_g=\{a_1,b_1,\ldots,a_g,b_g\}$ be the usual generating set for $\Gamma_g$ satisfying the surface ...
9
votes
0answers
188 views

Group with unsolvable conjugacy problem but solvable conjugacy length?

Could there exist a finitely presented group with unsolvable conjugacy problem, in which it is decidable whether a word over the group generators is a shortest representative of an element in its ...
0
votes
2answers
328 views

quotient groups of the lower central series of a free group

I have a question about some quotient groups of the lower central series of a free group. When there's a free group $F = \langle x_1,\cdots, x_n, y_1, \cdots, y_m\rangle $, let $A$ be the subgroup ...
1
vote
1answer
379 views

Dehn presentation of a knot group

The knot group is the fundamental group of the knot complement in $S^{3} $. The Dehn presentation of the knot group is a particular group presentation obtained by looking at the regions and crossings ...
16
votes
1answer
421 views

What is the length of the shortest law of $S_n$?

What is the length of the shortest word $w\in F_2$ such that $w(x,y)$ is trivial for every $x,y\in S_n$? There is a simple argument showing that we must have $\ell(w)\geq n$. See here for instance. ...
0
votes
0answers
202 views

Finitely presented group and its subgroups

Suppose I have a finitely presented group $G$. By this, I mean I know explicitly what $S$ and $R$ are such that $G = \langle S \mid R \rangle$. Suppose I have a subgroup generated by a finite set of ...
9
votes
1answer
249 views

Does every group embed into a co-hopfian group?

A group $G$ is co-hopfian if every injection $f\colon G \rightarrow G$ is an automorphism, or equivalently if $G$ is not isomorphic to any of its proper subgroups. Miller and Schupp, using small ...
13
votes
2answers
476 views

Kernel of linear representation of Baumslag-Solitar group

Let $BS(m,n)$ be the Baumslag-Solitar group defined by $B(m,n) = < a,b ~|~ b a^m b^{-1} = a^n > $, $mn \neq 0$. There is a linear representation of $BS(m,n)$ by mapping $a$ to the matrix ...
8
votes
1answer
243 views

The equality problem between conjugate group elements

The Novikov--Boone Theorem, which is perhaps the archetypal local unsolvability result in group theory, states existence of a finitely presented group whose word problem is recursively unsolvable. ...
2
votes
1answer
233 views

Presentations of infinite index subgroups

Suppose we have a finitely presented group $G$ with a concrete presentation and a subgroup $H$, generated by a finite set of elements from $G$. How to find the presentation for $H$? If $H$ has finite ...
0
votes
1answer
161 views

Monodromy in presentations of one group over another

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 ...
2
votes
1answer
427 views

Questions on the group with two generators $a,b$ and one relation $b^2=1$

Let $G$ be the finitely presented group with two generators $a,b$ and one relation $b^2=1$. First question: Does that group have a name ? Perhaps an answer to this question can lead me to ...
4
votes
2answers
298 views

Generating a group by randomly sampling generators

Let $G$ be a finite abelian group, $n$ a positive integer and let $G^n$ denote the direct product of $n$ copies of $G$. We say an element of $G^n$ is full if it acts as a nonidentity element of $G$ in ...
5
votes
0answers
202 views

Any method to detect subgroup generated by a subset of the generators from its presentation

I have met the following problem. A group $G$ is given as follows $G = \langle x,y,t| y^{-2}xy^2 = x,t^{-1}yt =y^2 ,t^{-1}xt = xy^{-1}xy\rangle$ Is the subgroup generated by $y$ and $t$ just the ...
3
votes
1answer
508 views

Cyclic subgroups of finite abelian groups

I learned from MO Subgroups of a finite abelian group that the problem of enumerating subgroups (not up to isomorphism) of finite abelian groups is a difficult one. Are there simple formulas if one ...
4
votes
3answers
413 views

Results in the Presentation of Finite Groups

I've been looking at combinatorial group theory, but all the results seem to be about infinite groups. Are there any important results about the presentations finite groups specifically (or are useful ...
1
vote
1answer
277 views

When $[G_k,G_m] = G_{k+m}$?

Hello? I have a simple question about combinatorial group theory. For a group $G$, I saw $[G_k, G_m] \subset G_{k+m}$ and these two subgroups need not be equal. Then is there any known condition that ...
1
vote
1answer
515 views

Any subgroup of f.g. free group with finite index contains a term of lower central series?

Hello? I have some questions in the group theory. I know that the intersection of the lower central series of a finitely generate free group is trivial. So I wonder whether every nontrivial subgroup ...
4
votes
1answer
763 views

For what finite groups is the cardinality of a minimal generating set well defined?

Recently I learned that the cardinality of a minimal set of generators of a finite $p$-group $G$ is well defined namely it is equal to the dimension of $H^1(G,\mathbb{F}_p)$. Moreover, if ...
3
votes
1answer
713 views

Reidemeister-Schreier Method for Finding Stabilizer of an Element in a Group Action

Hi, Everyone: I would appreciate some references for the version of Reidemeister-Schreier that is used to find the stabilizer of a point under a group action. The only refs. I have found are about ...
3
votes
1answer
384 views

Looking for deterministic criteria to generate the symmetric group?

So let $S_N$ be the symmetric group of degree $N$. We think of it as a permutation group via its natural action on the set $T=\{1,2,\ldots,N\}$. Say that $H\leq S_N$ is a subgroup which acts ...
4
votes
1answer
327 views

Reference request: discrete harmonic functions and ends of graphs

Let $G$ be an infinite locally finite connected graph with finitely many ends. A real-valued function $f : G \to \mathbb{R}$ is harmonic if $$f(v) = \frac{1}{d_v} \sum_{v \sim w} f(w)$$ where $v ...
13
votes
1answer
735 views

A synopsis of Adyan’s solution to the general Burnside problem?

Where can I find a high-level overview of Adyan’s original proof of the existence of finitely generated infinite groups with finite exponent? Additionally: If possible, would an expert ...
8
votes
1answer
515 views

Embedding of Baumslag-Solitar group into a certain group

Let $G$ be a group generated by $a_0, a_1, a_2$ with relations: $a_0 a_1 a_0^{-1}=a_1^4$ $a_1 a_2 a_1^{-1}=a_2^4$ $a_2 a_0 a_2^{-1}=a_0^4$ I am wondering if $BS(1,4)=\langle ...
1
vote
1answer
150 views

Reference request: lattice operations on the class of finitely presented groups

In my research, I work with certain finitely presented quotients of Coxeter groups. These are the automorphism groups of abstract polytopes, which are combinatorial generalizations of "usual" ...
16
votes
0answers
414 views

Density of first-order definable sets in a directed union of finite groups

This is a generalization of the following question by John Wiltshire-Gordon. Consider an inductive family of finite groups: $$ G_0 \hookrightarrow G_1 \hookrightarrow \ldots \hookrightarrow G_i ...
20
votes
3answers
930 views

In an inductive family of groups, does the probability that a particular word is satisfied converge?

We have some group word $w$ in $k$ letters. We say a $k$-tuple of group elements $\vec{g} = (g_1, g_2, \ldots , g_k) \in G^k$ satisfies the word $w$ if $w$ gives the identity at $\vec{g}$. More ...
10
votes
1answer
474 views

Finite-dimensional version of the word problem for groups

The (uniform) word problem for groups can be stated in several equivalent ways: Word Problem for Groups (WP) Instance: A finite presentation of a group G and an element w of G as a product of ...
5
votes
1answer
269 views

Can all terms of the Johnson filtration be hom-mapped onto the same nontrival group?

Let $F_n$ where $n \ge 3$ be a free group and let $(\mathcal A_n(k))$ where $k \ge 1$ be the kernel of the homomorphism $Aut(F_n) \to Aut(F_n/\gamma_{k+1}(F))$ determined by the natural homomorphism ...
10
votes
5answers
2k views

What can be said about a group from its presentation?

This maybe a very general question. If we have a group given by its presentation only, what kind of properties could be proven about it? I know examples about non-amenability of some Burnside ...
15
votes
4answers
754 views

Free splittings of one-relator groups

Roughly speaking, I want to know whether one-relator groups only have 'obvious' free splittings. Consider a one-relator group $G=F/\langle\langle r\rangle\rangle$, where $F$ is a free group. Is ...