# Questions tagged [combinatorial-group-theory]

The combinatorial-group-theory tag has no usage guidance.

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### Darkness in the lamplighter group

Consider paths through the lamplighter group $\mathbb{Z}_n\wr\mathbb{Z}$ with steps consisting of moving left, moving right, and toggling the lamp at the current position. How many paths of length $m$ ...

**5**

votes

**3**answers

147 views

### Conjugacy in right-angled Artin groups

I am looking for a reference containing the following result:
Let $a$ and $b$ be two elements of a right-angled Artin group $A$. Assume that $a$ and $b$ have minimal length (with respect to the ...

**11**

votes

**1**answer

386 views

### Is there a name of semidirect product of a group with its automorphism group?

Consider the construction $G \rtimes \text{Aut}(G)$. Here $
G$ is a group, $\text{Aut}(G)$ is the automorphism group and the semidirect product is over the most obvious action.
1) Is there any name ...

**3**

votes

**1**answer

160 views

### When is the semidirect product of $(Z/pZ)^n$ and $(Z/qZ)^2$ generated by two elements?

I asked a very similar question here and got a wonderful answer. But now I need to change the question slightly (this is the last question like this, I promise).
I would like to characterize when $(\...

**9**

votes

**1**answer

319 views

### When is the semidirect product of an elementary abelian group and a cyclic group generated by two elements?

I am trying to characterize when a semi-direct product of the form $(Z/pZ)^n \rtimes (Z/qZ)$ is isomorphic to a group generated by two elements. Here $p$ and $q$ are distinct odd primes.
I would be ...

**10**

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**0**answers

276 views

### Amalgamated product of automatic groups

In Gersten's "Problems on Automatic Groups", Problem 14, he asks the following question: Let $G=A\ast_{C}B$ where $A$ and $B$ are automatic and $C$ is infinite cyclic. Is $G$ automatic?
Is this ...

**2**

votes

**1**answer

194 views

### Quotient groups of the lower central series of a surface group

In the answer to MO question 132247, it is possible to find a nice computation of the quotient groups of the lower central series of a finitely generated free group.
Q. What are the quotient ...

**19**

votes

**1**answer

430 views

### Can a hyperbolic, one ended, one relator group, have a shorter trivial word?

Let $G= \langle S \mid r \rangle$ be a one-relator presentation for a one-ended hyperbolic group, with $r$ cyclically reduced.
Question: Can there be a nontrivial word $w(S)$ which is trivial in the ...

**6**

votes

**1**answer

379 views

### Minimum number of operations necessary to arrive at any configuration

Let $k \geq 2$ and $N_1, N_2, ..., N_k$ be positive integers.
Let $S=\{(a_1,a_2,...,a_k) \in \mathbb{Z}^k:1 \leq a_i \leq N_i\}$ and $A=\{1,2,...,\prod_{i=1}^{k} N_{i}\}$.
Given a bijective map $f:...

**10**

votes

**1**answer

139 views

### Minimal length presentations of cyclic groups

By the length of a finite presentation I mean the sum of the lengths of the relators. I am interested in knowing what the minimal length of a presentation of $\mathbb{Z}/n\mathbb{Z}$. I'm even more ...

**6**

votes

**1**answer

314 views

### An algorithm determining whether two subgroups of a finitely generated free group are automorphic

In the book Lyndon, Schupp, Combinatorial Group Theory, P.30 in the edition from 2000 They mention an unpublished work by Waldhausen that is said to give an algorithm to determine whether two ...

**10**

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**0**answers

117 views

### 2-generator subgroups of an Artin group of small type

Suppose I have an Artin group $G$ of small-type, meaning that the generators either commute or braid. E.g a braid group. Take two generators $g, h$ and arbitrary conjugates of these generators $xgx^{-...

**11**

votes

**1**answer

250 views

### “Bisecting” a free subgroup with respect to word length

My broad question is regarding the lengths of (reduced) words in a subgroup of a free group.
As motivation, consider the free group $Gp(S)$ where $|S|=n$, that is, a free group of rank $n$. Let $S=\{...

**3**

votes

**1**answer

101 views

### Maximal power in a sequence of iterated commutators in the rank two free group

I have the following problem: in the free group $F_2=\langle a,b\rangle$, we define the sequence
$\begin{cases}
w_0=a, \\
w_1=b, \\
w_{n+2}=[w_{n+1},w_{n}] & \text{for }n\ge 0.
\end{cases}$
So $...

**2**

votes

**0**answers

118 views

### Concentration of Reduced words

This might be a rather broad question, and I'll be satisfied with some intuition and pointers to relevant literature. However, I'll certainly fill in more context and details based on any feedback.
...

**2**

votes

**2**answers

263 views

### Subgroup of a free group that is characteristic but not totally characteristic

Looking for a counter example (if it exists) and a reference for further reading. Can there be a subgroup of finite index in a finitely generated free group that is characteristic but not totally ...

**2**

votes

**0**answers

86 views

### Change of generators and shortest product in groups

Let $G$ be a finitely generated group.
For a set of generators $B$ of $G$, $\ell_B(x)$ is the length of the smallest sequence of elements(and inverse of the elements) in $B$, such that the product ...

**3**

votes

**2**answers

203 views

### Free subgroup of a quotient

Let $F$ be a free group on $x,y,z$. Fix $n>1$ (I am ready to assume that $n$ is large enough). Let $\mathcal{W}$ be the set of cyclically reduced words $w$ in $F$ where the letter $z$ appears at ...

**4**

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**0**answers

93 views

### Order problem in nilpotent groups

Let $G$ be a f.g. nilpotent group. I wanted to know if the order problem (given $g \in G$, deciding if there exists $n$ s.t. $g^n=e$) is decidable in $G$? In such a group, the word problem is ...

**8**

votes

**1**answer

231 views

### Need a good name for an algorithmic problem in groups that generalizes the conjugacy problem

I am looking for a good name for the following problem:
Given elements $g_1,\dotsc,g_n$ in a (finitely generated) group $G$, determine if the product of their conjugacy classes $g_1^G\dotsb g_n^G$ ...

**0**

votes

**0**answers

90 views

### Expected shortest word length\depth in Braid groups from set of all braids with length L

Consider the set of all positive braids on n strands, with a fixed length L.
$$B^+_{n,L}:=\{\beta\in B_n:\beta=\sigma_{i_1}\sigma_{i_2}\ldots\sigma_{i_L},1\leq i_k \leq n-1\}$$.
Using the relations,
...

**2**

votes

**0**answers

186 views

### When can we establish an isomorphism between two not-finitely presented groups?

Assume that finitely generated groups $G$ and $H$, are not finitely presented. Fix a generating set $\mathfrak g:=\{g_1,\dotsc,g_n\}$ of $G$. Let $\mathfrak R:=\{R_1,R_2,\dots\}$ be the set of all ...

**2**

votes

**1**answer

150 views

### Could the number of defining relators of a finitely presented group increase

Do there exist finitely generated groups $G$ and $H$ with following properies:
$G=\langle g_1,\dotsc,g_n\rangle$ is not finitely presented, Let $R:=\{r_1,r_2,\dots\}$ be the set of its defining ...

**4**

votes

**2**answers

395 views

### A question about generating set of groups and epimorphism

Do there exist non-isomorphic finitely generated groups, $G$ and $H$, along with epimorphisms $\phi:G\rightarrow H$ and $\psi:H\rightarrow G$, such that every generating set of these groups is an ...

**8**

votes

**1**answer

204 views

### Does every generating set of the first homology group of a Cayley graph give rise to a presentation of its group?

Let $G$ be a group, and fix a symmetric generating set $S$. Let $X$ be the corresponding Cayley graph.
Let $R$ be a set of words in $S$, each corresponding to the identity of $G$, such that the set ...

**21**

votes

**1**answer

930 views

### Covering of a surface of a cube $n\times n \times n$ by pieces of paper $1\times 6$

When I was too young one of my problems was in the list of problems of All-Russian Olympiad. The problem is the following:
Problem. We have a surface of a cube $n\times n \times n$ such that each ...

**4**

votes

**1**answer

119 views

### Computations with conetypes of hyperbolic groups

I'd like to know if there exists (and, in this case, where I can find it) some computer program/programming language/any kind of software that can find explicitly the conetypes of a hyperbolic group ...

**5**

votes

**1**answer

279 views

### Is there a one relator group with property (T)?

Is there an $n > 2$, and some $x \in F_n$ (the free group on $n$ generators) such that the quotient of $F_n$ by the normal subgroup generated by $x$ has Kazhdan's property $\mathrm{(T)}$ ?

**0**

votes

**1**answer

86 views

### Rank of a generall linear group over a finite field [closed]

What is the rank (minimal number of group generators) of the group $GL(n,F)$, when $F$ is a finite field of odd order? I found that $SL(n,F)$ is $2$, but I can't find this information.

**1**

vote

**4**answers

269 views

### Are the orders of the generators of a group and the product of pairs of thereof enough for this group to be isomorphic to a Coxeter group?

Let's say we have $n$ generators $x_1, x_2, \cdots, x_n$ along with the following facts concerning their orders:
\begin{eqnarray*}
ord(x_i) &=& 2 \text{ for } i = 1, 2, \cdots, n \\
ord(x_i ...

**3**

votes

**3**answers

123 views

### Are there references for the properties of words formed in finite groups using L-systems? (In particular, the algae L-system.)

Let $G$ be a (finite) group, and $a, b \in G$ be any two elements. Consider the sequence defined by
\begin{eqnarray*}
s_0 &=& a, \\
s_1 &=& b, \text{and} \\
s_{n+2} &=& s_{n+1} ...

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votes

**1**answer

560 views

### Avoiding countable subgroups of a group homeomorphic to the Cantor space

Update: Further work with Adam (who answers below) and Piotr led to a rather satisfactory result about the problem that motivated the problem below, see our recent paper
The Haar Measure Problem. In ...

**8**

votes

**2**answers

638 views

### Avoiding countable subgroups of general uncountable groups

The following problem is a general form of another problem (motivation is available there). Initially, the problems were posted together, but the first one is solved below, a solution that does not ...

**5**

votes

**1**answer

291 views

### K-fellow traveler property and automatic structure

I have been reading several articles about automatic groups and metric spaces of negative curvature. However it is not clear for me the relationship between automatic groups, hyperbolcity and the k-...

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votes

**3**answers

464 views

### Rank of a special linear group over a finite field

What is the rank (minimal number of group generators) of $SL(n,\mathbb{F})$ in the situation when $SL(n,\mathbb{F})$ is not perfect (i.e. when $SL(n,\mathbb{F})$ is different from $SL(2,\mathbb{F}_2)$ ...

**1**

vote

**0**answers

117 views

### Intuitive meaning of benign subgroup

Disclaimer! This is a copy of a question I posted on M.SE!
I still think the question belongs there but I'm not getting any answers so I'm dublicating with slight changes:
I've been studying a proof ...

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votes

**2**answers

507 views

### Why is “The Higman Rope Trick” thus named?

I'm studiyng Higman's Embedding Theorem, and a fundamental part of the proof is the following lemma:
If R is a benign normal subgroup of finitely generated group F, then F/R can be embedded in a ...

**3**

votes

**1**answer

250 views

### Reference request for non-commutative analogues of exterior algebras

I am reading Combinatorial Group Theory In Homotopy Theory I by Fred Cohen (preprint available on web page). Here is an extract of the paper:
Cohen called $A^R_n$ "a standard tool used in ...

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votes

**1**answer

455 views

### Paper by I. N. Sanov, Solution of the Burnside problem for exponent 4

I have searched extensively online and for copies of printed journals containing the paper which details Sanov's solution to the Burnside Problem for exponent 4, which is widely cited in many papers ...

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vote

**1**answer

156 views

### Request a paper by Fred Cohen

I am looking for the following paper by Cohen, F. R.:
On combinatorial group theory in homotopy. Homotopy theory and its
applications (Cocoyoc, 1993), 57–63, Contemp. Math., 188, Amer. Math.
...

**4**

votes

**0**answers

159 views

### Log-concavity of the growth function

Given a Cayley graph of a group $G$ with finite generating set $A$ and exponential growth. Let $S_n$ be the elements whose word length is exactly $n$.
$\textbf{Question:}$ Is $f(n) = |S_{2n}|$ a log-...

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votes

**2**answers

660 views

### Which groups are LERF?

A finitely generated group $G$ is called LERF if every finitely generated $H \leq G$ is closed in the profinite topology on $G$ (equivalently, there is a family of finite index subgroups of $G$ ...

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votes

**0**answers

267 views

### Marshall Hall's theorem for surface groups [closed]

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

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votes

**2**answers

304 views

### 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 ...

**5**

votes

**1**answer

224 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, ...

**0**

votes

**1**answer

66 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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133 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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votes

**2**answers

272 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

**1**answer

260 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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votes

**0**answers

4k 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.