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Questions about the branch of algebra that deals with groups.

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Some questions on a paper of Baumslag and Solitar

I was recently trying taking a look at the paper "Some two-generator one-relator non-hopfian groups" by Baumslag and Solitar where they introduce the groups now known as the Baumslag-Solitar ...
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Intersection of $\mathrm{PGL}_2(q_0)$'s in $\mathrm{PGL}_2(q_0^3)$

$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\GL{GL}$I would like an explanation for some strange numerology which I encountered when studying intersections of subfield subgroups in $\PGL_2(q)$. ...
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1answer
272 views

A detail in the proof of Schur's lemma: the closures of the $\mathcal{Ker}$ and $\mathcal{Im}$ of the intertwiner

$\renewcommand\Im{\operatorname{\mathcal{Im}}}\newcommand\Ker{\operatorname{\mathcal{Ker}}}$I was sure that this is a trivial question and placed it on Math Stackexchange https://math.stackexchange....
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1answer
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The $\{2,3\}$-groups with a condition about $\mathbb{C}$-characters

Let $G$ be a $\{2,3\}$-group and $\lvert G\rvert=2^\alpha\cdot3^\beta$. For $p\in\{2,3\}$, define $$ \nu_p(G)\mathrel{:=}\min\left\{\log_p\left(\frac{\lvert G\rvert}{\chi(1)}\right)_p \mathrel{\...
8
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1answer
172 views

Geometric intuition behind Garside's paper?

I apologize in advance for a somewhat wishy-washy question. I just read the paper "The Braid Group and Other Groups" by F. A. Garside in which he solves the conjugacy problem for the braid ...
3
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49 views

Classification of maximal point groups

Have the maximal (without finite proper overgroups of the same dimensionality) finite point groups been fully classified in any dimensionality of Euclidean space greater than 4?
5
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1answer
139 views

Number of subgroups of a $p$-group of index $p^k$

Let $p$ be a prime, let $n$ and $k$ be positive integers and let $G$ be a group of order $p^n$. Further, let $a_{p^k}$ denote the number of subgroups of $G$ of index $p^k$. If $a_{p^k}$ is greater ...
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Computability of the “free envelope rank” of an endomorphism of a free group

Let $F$ be a free group freely generated by the finite set $S$ and $\sigma\colon F\to F$ be a group morphism. We define the free envelope rank of $\sigma$, written $r(\sigma)$, as the smallest $k$ for ...
6
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1answer
211 views

Swapping non-commuting generators in Coxeter group

Let $a$ and $b$ be two generators in a Coxeter group which do not commute. Is it possible for $ab$ to be equal to a product of generators where all instances of $b$ come before all instances of $a$? I'...
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227 views

Properties of C′(1/6) groups

I asked this question on Math Stack Exchange about a week ago, but nobody answered. Let $G = \langle X \mid R\rangle$ be a group presentation, where $R$ \subset $F(X)$ is a set of freely and ...
12
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1answer
270 views

Commutator problem vs conjugacy/word problem

For a finitely presented group $G$, generated by a finite set $A$, the commutator problem is the decision problem: given a word $w$ over the alphabet $A \cup A^{-1}$, can one decide if $w$ is a ...
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Is this equivariant function constant?

Let $G$ be a linear algebraic group (think of $SL_n(\mathbb{R})$), $B$ its Borel (standard minimal parabolic) subgroup (think of upper triangular subgroup), and let $\Gamma \leq G$ be a cocompact ...
4
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1answer
150 views

Finite covolume of uniform lattice in quotient group

Let $G$ be a locally compact group, let $N \leq G$ be a (proper) closed normal subgroup and let $\Gamma \leq G$ be a uniform lattice, i.e., a discrete subgroup such that $G/\Gamma$ Is compact. Suppose ...
5
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1answer
223 views

Quotient groups obtained by quotienting $G^n$ by $G^{n-1}$

Notation: For a group $G$, we write $G^n$ to denote the $n$-fold direct product of $G$ with itself. Problem set up: Consider, for a finite group $G$, and $n > 1$, the set $Q(G)_n$ of all ...
5
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243 views

Is there a higher categorical structure which models the (higher) conjugation actions of a group acting on itself?

Let $G$ be a group, and consider the action of $G$ on itself by conjugation. If we think of $G$ as a one object category, then the conjugation action can be realised as automorphisms of this category, ...
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81 views

Large subgroups of infinite-dimensional vector spaces

Let $V$ be an infinite-dimensional vector space over $\mathbb{Q}$. Consider a proper subgroup $W$ of $V, +$ with the following property: each vector line $L$ (which we see as a subgroup of $V, +$) has ...
4
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4answers
456 views

The integral closure $\overline{\mathbb{Z}}$ and the group $\overline{\mathbb{Z}}^{\times}$

Let $\mathbb{Q}$ be the field of rational numbers, and let $\overline{\mathbb{Q}}$ be its algebraic closure. Assume $\overline{\mathbb{Z}}$ is the integral closure of $\mathbb{Z}$ in $\overline{\...
9
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1answer
148 views

Group extensions with non-abelian kernel

If $N$ is a normal subgroup of $G$ then there is a coupling: that is, a representation of $G/N$ in $\operatorname{Out}(N)$. In that case, the extensions of $N$ by $G/N$ affording the same coupling are ...
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94 views

Restrictions on pointed lifts of isometries

Let $M$ be a (closed) Riemannian manifold and let $f$ be an isometry of $M$ fixing a point $\ast \in M$ that acts trivially on $\Gamma := \pi_1(M,\ast)$. Then there is a unique isometry $\tilde{f}$ of ...
2
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1answer
123 views

A variation of closed-subgroup theorem

$\DeclareMathOperator\SO{SO}$Recall that the closed-subgroup theorem (Wikipedia link) says that a closed subgroup of a Lie group is a Lie group. I am pretty sure that this theorem should have a "...
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39 views

Ideal Ford domain (for finite index subgroup)

Let $G$ be a lattice Fuchsian group with parabolic elements, seen as a discrete subgroup of matrices $ g= \begin{pmatrix} \alpha & \overline{\beta} \\ \beta & \overline{\alpha} \end{pmatrix} $...
6
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1answer
185 views

Is the function $k(g,h) = \frac{1}{1+\lvert gh^{-1}\rvert}$ positive definite?

Let $G$ be a finite group, $S \subset G$ a generating set, closed under taking inverses, and $\lvert\cdot\rvert$ the word length with respect to this set $S$. Question. Is the function $k(g,h) = \...
3
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0answers
96 views

Ping pong with parabolic isometries on Gromov hyperbolic spaces

For a group $G$ with a non-elementary general type action by isometries on a Gromov hyperbolic geodesic space $(X,d)$, it is well known that you can construct free subgroups of $G$ via the ping pong ...
1
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1answer
128 views

Subgroup rank of finite simple groups

Definition: The subgroup rank of a finite group $G$ is the minimal natural number $n$ such that every subgroup of $G$ can be generated by $n$ elements (or fewer). This invariant has been studied ...
2
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0answers
62 views

Weights of finite abelian group actions on submanifolds/subvarieties

(cross-posted from https://math.stackexchange.com/questions/4125529/weights-of-finite-abelian-group-actions-on-submanifolds-subvarieties) How do weights associated to actions of finite subgroups of $\...
3
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1answer
69 views

Example of a primitive group of affine type and of twisted wreath product type

Good evening, According to the O'Nan-Scott theorem, primitive finite group are classified into five classes: affine type, product type, almost simple type, diagonal type and twisted wreath product ...
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1answer
149 views

On the coefficients that appear in finite groups of matrices with integer entries

Let $n$ be a positive integer and $G$ be a finite group of $n\times n$ matrices with integer coefficients, i.e. $G\subset\operatorname{GL}_n(\mathbb{Z})$. It is known that for sufficiently large $n$, ...
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1answer
284 views

Are there overwhelmingly more finite monoids than finite spaces? [closed]

A function $f:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ overwhelms $g:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ if for any $k\in \mathbb{Z}_{\geq 1}$ the inequality $f(n)\leq g(n+k)$ holds only for ...
3
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1answer
373 views

Are there overwhelmingly more finite posets than finite groups? [closed]

A function $f:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ overwhelms $g:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ if for any $k\in \mathbb{Z}_{\geq 1}$ the inequality $f(n)\leq g(n+k)$ holds only for ...
5
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0answers
189 views

Terminology question in group actions

Given a continuous group action $G \times X \rightarrow X$ on a topological space $X$, is there a standard term for the subsets $K \subset X$ for which Every open neighborhood of $K$ intersects every ...
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166 views

Hopficity of Baumslag-Solitar groups

I am struggling to find the exact source (with proofs) of the following ''well-known'' statement: the Baumslag-Solitar group $BS(m,n)=\langle a,t \mid ta^m t^{-1}=a^n\rangle$ is Hopfian if and only if ...
7
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1answer
199 views

Fusing conjugacy classes

Consider a finite group $G$ and two conjugacy classes $H$ and $I$ of isomorphic subgroups of $G$. Question. Is there some finite overgroup of $G$ which fuses $H$ and $I$ into a single conjugacy class?...
7
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0answers
102 views

Can a lacunary hyperbolic group be Liouville?

While discussing with a colleague of a possible use of lacunary hyperbolic elementarily amenable groups introduced by Olshanskii, Osin & Sapir, it occurred to me that I was not aware whether ...
7
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2answers
221 views

Index of the mapping class group $\Gamma_{g,n}$ inside $\text{Out}(\Pi_{g,n})$

Let $\Sigma_{g,n}$ denote an $n$-punctured surface of genus $g$, with $2g+n-2 > 0$. Let $\Pi_{g,n}$ be its fundamental group (for some choice of base point), and let $\Gamma_{g,n}$ denote its pure, ...
3
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98 views

Torsion-free subgroups in arbitrary lattices

Let $\Gamma $ be a lattice in a semi-simple Lie group $G$. If the Lie group $G$ is linear (that is, it has a faithful finite-dimensional linear representation), then $\Gamma$ contains a torsion-free ...
8
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1answer
375 views

Definition of an arithmetic subgroup of an algebraic group

I'm struggling with the definition of an arithmetic subgroup of an algebraic group defined over $\mathbb{Q}$. In Wikipedia you can read: If $\mathrm G$ is an algebraic subgroup of $\mathrm{GL}_n(\...
7
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1answer
306 views

Independent vectors in the permuting coordinates action of $S_n$ on $\mathbb{R}^n$

Let $V$ be the hyperplane in $\mathbb{R}^n$ with equation $\sum_i x_i=0$. The symmetric group $S_n$ acts on $V$ by $s\cdot (v_1,\ldots,v_n)=(v_{s^{-1}(1)},\ldots,v_{s^{-1}(n)})$. Consider those $v\in ...
4
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1answer
91 views

Computable change in minimum word length of subgroup elements

Let $G$ be an infinite finitely generated group. Fix a finite generating set for $G$. Define $\mathrm{len}_G:G\to\mathbb{Z}_{\geq 0}$ by sending $g$ to the minimum length of a word in the generators ...
7
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1answer
200 views

Are two quasi-isometric, isomorphic on large enough balls, transitive graphs isomorphic?

Take two transitive graphs $X,Y$ (potentially directed and edge-labelled, e.g. Cayley graphs). Assume $X,Y$ are quasi-isometric with constant $K$, i.e. there exists a function $f:VX \to VY$ ($VX,\,VY$ ...
4
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0answers
91 views

Finitely presented group with exactly two conjugacy classes

Is there a finitely presented group with exactly two conjugacy classes other than $\mathbb{Z}/2\mathbb{Z}$?
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64 views

A criterion for loxodromicity in Gromov-hyperbolic spaces

Recall that an isometry of a Gromov-hyperbolic space $X$ is called loxodromic if it has exactly two fixed points on the Gromov boundary $\partial X$, one being "attracting" and the other &...
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98 views

Decidable membership for subgroup generated by three elements in $F_2\times F_2$

Let $F_2$ be the non-abelian free group on two generators. Let $G\subset F_2\times F_2$ be a subgroup generated by three elements. Is there an algorithm deciding if a given element of $F_2\times F_2$ ...
7
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1answer
115 views

Infinite oscillation of minimum word length in 2-generated group

Let $G$ be a group with generators $a, b\in G$. Define $\mathrm{len}:G\to\mathbb{Z}_{\geq 0}$ by sending $g$ to the minimum length of a word in $a, b, a^{-1}, b^{-1}$ equal to $g$. Assume that for all ...
2
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1answer
273 views

Is the representation of finite simple groups fully understood?

Is the representation of finite simple groups fully understood? To clarify, I mean have all the simple representations (even finite dimensional) been classified in terms of some classifying set, such ...
0
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0answers
90 views

How to classify rings by combinatorial structures?

There are many ways to encode information about algebraic structures such as groups, rings, etc... in combinatorial form. For example the Cayley graph of a group with a subset of generators, or the ...
7
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3answers
355 views

Membership to double cosets in free groups

Is there an elementary and efficient algorithm for testing the membership to a double coset of f.g. subgroups in a free group? Has this membership problem been implemented in GAP/Magma? More ...
2
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0answers
62 views

Terminology and notation for generated subgroups

I would like to think about formation of the smallest subgroup (or monoid, or whatever) $H$ of $G$ containing two given subgroups $A$ and $B$ as an operation on subgroups, and I wonder if there is a ...
7
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1answer
233 views

Induction and restriction of unitary representations

$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Ind{Ind}\DeclareMathOperator\Res{Res}$Given a locally compact group $G$ and a closed subgroup $H\subset G$, let $\Rep(G)$ and $\Rep(H)$ denote their ...
6
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0answers
131 views

Is it possible for a finitely generated group to have polynomial growth rate with leading coefficient less than 1?

I'm quite a neophyte in this area, so apologies if this question is easy. (I've edited slightly to make my question more clear.) I've been trying to learn about growth rates for finitely generated ...
5
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1answer
65 views

Equality to a power of a given word undecidable in finitely presented group with decidable word problem

Let $G$ be a group with an explicit finite presentation. Assume $G$ has a decidable word problem. Can there exist an explicit word $w\in G$ such that there is no algorithm deciding if a given word $w'\...

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