The combinatorial-group-theor tag has no wiki summary.

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### A Karrass-Solitar theorem for surface groups

Let $\Gamma_g$ be a surface group of genus $g \geq 2$. That is, there is a presentation $$\Gamma_g = \langle x_1, y_1, \dots, x_g, y_g \vert \prod_{i = 1}^{g}[x_i,y_i] = 1\rangle$$
Is there a ...

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**1**answer

157 views

### Bases of surface groups

Let $\Gamma_g$ be a surface group of genus $g \geq 2$. A $2g$-tuple $(x_1,y_1, \dots,x_g,y_g) \in \Gamma_g^{2g}$ will be called a Surface Basis if we have the presentation $$\Gamma_g = \langle x_1, ...

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**1**answer

53 views

### Monotonicity of the gap of permutated sequence

Let $a$ be an arbitrary sequence and denote by $\mbox{gap}_k(a) = a_{(k)} - a_{(k+1)}$, where $a_{(k)}$ is the $k$th largest component of $a$. Of course, $k+1$ should be no larger than the length of ...

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93 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 ...

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184 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 ...

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votes

**1**answer

179 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 ...

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383 views

### How many combinations does Android pattern have? [closed]

Rules-
1) At-least 4 and at-max 9 dots must be connected.
2) There can be no jumps
3) Once a dot is crossed, you can jump over it.

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**1**answer

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 ...

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votes

**1**answer

135 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 ...

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351 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 ...

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**0**answers

189 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 ...

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vote

**1**answer

181 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 ...

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321 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 ...

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**1**answer

245 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 ...

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**1**answer

404 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 ...

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191 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 ...

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**2**answers

347 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 ...

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vote

**1**answer

395 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 ...

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**1**answer

430 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. ...

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210 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 ...

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**1**answer

252 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 ...

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479 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 ...

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249 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. ...

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**1**answer

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 ...

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**1**answer

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 ...

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**1**answer

431 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 ...

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299 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 ...

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203 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 ...

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**1**answer

511 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 ...

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418 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 ...

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262 views

### question about derived subgroup

Let $G$ be a free group. Then $G/G^{(n)}$ ($G^{(n)}$ is the $n$th derived subgroup.) acts on $G^{(n)}/G^{(n+1)}$ by conjugation, which makes $G^{(n)}/G^{(n+1)}$ a $\mathbb{Z}[G/G^{(n)}]$-module. What ...

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**1**answer

536 views

### Minimal generation for finite abelian groups

Let $G$ be a finite abelian group. I know of two ways of writing it as a direct sum of cyclic groups:
1) With orders $d_1, d_2, \ldots, d_k$ in such a way that $d_i|d_{i+1}$,
2) With orders that are ...

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**1**answer

279 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 ...

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**2**answers

242 views

### In hyperbolic 3-orbifold with totally geodesic boundary case, is it true: rank(the fundamental group of boundary M)< or equal 2 rank(fundmental group of M)?

For a orientable three manifold M with totally geodesic boundary, this inequality is true. Because the rank of (fundemantal group of boundary M)=rank (homology group of boundary M )
then we use the ...

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**1**answer

103 views

### Will the rank of fundemantal 3 manifold be decreased is I module the n(n>1) times of a element?

I am doing some rank control about the fundamental group of a 3 dim hyperbolic orbifold. After cuting out the regular neighborhood of all the singularity, I get a manifold with imcompressible ...

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**1**answer

517 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 ...

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votes

**1**answer

768 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
...

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602 views

### infinite group that maps onto dihedral group

The group is generated by $y_i$, $i=0, ...,p-1$
with relations
$y_0y_1=y_1y_2=...=y_{p-1}y_0$
$y_0y_2=y_1y_3=...=y_{p-1}y_1$
$\vdots$
$y_0y_{p-1}=y_1y_0=...y_{p-1}y_{p-2}$
I have run into this ...

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**1**answer

722 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 ...

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**1**answer

385 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 ...

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**2**answers

416 views

### Is every virtual knot group an HNN extension?

A basic fact in knot theory is that a knot group $\pi(K)$ is an HNN extension of $\pi(F)$, the fundamental group of a Seifert surface complement. A nice discussion of this may be found in Chapter 11 ...

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**1**answer

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 ...

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747 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 ...

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517 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 ...

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**1**answer

153 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" ...

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420 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 ...

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**1**answer

298 views

### What is the relative weight filtration of the mapping class group of a surface?

It is my understanding that Dennis Johnson defined a `relative weight filtration,' of the mapping class group of an oriented surface. My question is what is this filtration, and how does it relate to ...

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935 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 ...

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477 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 ...

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**1**answer

272 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 ...