Large scale properties of groups; growth functions; Dehn functions; small cancellation properties; hyperbolicity and CAT(0); actions and representations; combinatorial group theory; presentations

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0
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1answer
19 views

Double coset separability and the existence of vanishing sequences for surface group

Definition: Let $G$ be a group. $G$ is said to be double coset separable if given any finitely generated subgroups $H$ and $K$ in $G$, given any $g\in G$ and $h\not\in HgK$, there exists a finite ...
0
votes
0answers
53 views

One question about iteration on groups

Let $G$ be a finite generated group, $H$ a subgroup of $G$ of index $n$, with $a_i$ a set of coset representatives and $$G=\displaystyle\bigcup_{i=1}^nH{a_i}.$$ Let $\phi:H\rightarrow G$ be a ...
6
votes
1answer
150 views

Isometries of some simple Cayley graphs

Consider a Cayley graph of a group $G$ with respect to a symmetric finite generating set $S$. There are some obvious candidates to isometries of this graph - for example, translation by elements of ...
7
votes
1answer
204 views

Can you decide whether the commutator subgroup of a f.p. group is f.g?

Is the following algorithmic problem known to be decidable/undecidable? Input: a finite group presentation $P$. Decide: is the commutator subgroup of the group presented by $P$ finitely generated?
-4
votes
0answers
86 views

How to use Holder's Inequality to prove two Banach spaces are equal $(\mathbb R^n,\|\cdot\|_3)*= (\mathbb R^n,\|\cdot\|1.5)$ [closed]

How can one prove that Banach spaces $(\mathbb R^n,\|\cdot\|_3)*= (\mathbb R^n,\|\cdot\|1.5)$
0
votes
2answers
113 views

Are there CAT(-1) spaces which are not trees whose Gromov boundary is disconnected?

Are there some examples of CAT(-1) spaces which are not trees which have disconnected Gromov boundary?
-1
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1answer
186 views

Suppose that $G$ is a subgroup of $GL_n(\mathbb C)$ with finite exponent. Then is $G$ a finite group? [closed]

As title. the exponent of $G$ is the least number $n$ (if exists) such that $g^n=e$ holds for all $g\in G$ or $+\infty$.
3
votes
0answers
170 views

every element with eigenvalue 1

Is there a "non-trivial" subgroup $G$ of $GL(n,\mathbb C)$ whose every element has an eigenvalue $1$? "Non-trivial" here means that is no common eigenvector with eigenvalue $1$ for all elements of ...
1
vote
0answers
97 views

Discontinuity of Radon-Nikodym derivative for Patterson-Sullivan measures for word metrics on Gromov hyperbolic groups

Let $\Gamma$ be a Gromov hyperbolic group coming endowed with a word metric coming from some finite generating set. Let $\nu$ be an associated Patterson-Sullivan measure (quasi-conformal density). I ...
6
votes
0answers
170 views

When is a word metric on a CAT(-1) group a bounded distance from the orbit map of an isometric action on some CAT(-k) metric space?

Let $\Gamma$ be a group admitting a discrete and cocompact action on a CAT(-1) space. Let $d$ a word metric on $\Gamma$ coming from some finite set of generators. My question is: Does there exist a ...
18
votes
6answers
1k views

Residual finiteness: why do we care?

Residually finite groups have been studied for a long time. However, I am struggling to work out why we care, or perhaps, why they continue to be of interest. Let me explain. Magnus, in his 1968 ...
6
votes
0answers
167 views

Polynomial growth without Gromov's theorem

It is a consequence of the theorem of M. Gromov on groups with polynomial growth and a result of P. Pansu(*) that if a finitely generated group $\Gamma = \langle S\rangle,\, S=S^{-1}$ satisfies ...
13
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0answers
266 views

Linear groups which don't contain products of free groups

Let $G \subset GL(n, \Bbb Z)$ be a f.g. linear group. The Tits alternative says that $G$ is either virtually solvable (i.e. has a solvable subgroup of finite index), or contains a free group $F_2$. ...
1
vote
0answers
101 views

Virtually abelian centralizers

This is a sort of a follow-up question to Limits of conjugated subgroups (though it might not seem at first glance to have much to do with it.) Anyway, I'm wondering what sort of groups have the ...
8
votes
1answer
165 views

Is the Cayley graph of Thompson's group isolated in the space of vertex-transitive graphs?

Consider Thompson's group (the one commonly referred to as $T$), which is finitely presentable. Consider the Cayley graph, but then forget the coloring and direction on edges. So now we just have an ...
7
votes
0answers
64 views

Approximation of the effective resistance on Cayley graph

Let $\Gamma$ be a finitely generated group, and denote by $G$ the Cayley graph of $\Gamma$. Denote by $d_R$ the resistance distance metric on this graph. The resistance distance metric between the ...
4
votes
0answers
83 views

Is a finitely generated subgroup of a free profinite group virtually a retract?

Let $F$ be a nonabelian finitely generated free profinite group, and let $H \leq F$ be a finitely generated closed subgroup. Must there be some open subgroup $H \leq U \leq F$, and a closed normal ...
3
votes
0answers
70 views

Geometric automorphism of free group respect to nonorientable suface

An outer automorphism $[ϕ]\in Out(F_n)$ is geometric if it is induced by a surface homeomorphism h:M→M, where M is a compact surface with nonempty boundary. I am wondering is it enough we only ...
3
votes
1answer
179 views

Special linear groups over function fields

Let $p$ be a prime number, and let $q$ be a finite power of $p$. Denote by $F_q$ the unique field with $q$ elements. What is known about the structure and properties of $\mathrm{SL}_2(F_q[t])$ as ...
3
votes
2answers
190 views

Limits of conjugated subgroups

I've recently encountered the following problem. Given a group $G$, a subgroup $H$ and a sequence $g_n\in G$, let $$ \liminf_{j\to\infty}H^{g_j} :=\bigcup_{n\ge 1} \bigcap_{j\ge n} H^{g_j}.$$ Here $$ ...
1
vote
1answer
91 views

Convex subcomplexes of CAT(0) cubical complexes

Is the following statement true? If so, can anyone provide a reference? Let $X$ be a CAT(0) cubical complex, and let $Y$ be a connected subcomplex of $X$. Then the following are equivalent: ...
11
votes
1answer
340 views

Is there a faithful transitive locally finite action of the modular group?

Is there a faithful transitive action of $G = \mathrm{PSL}_2(\mathbb{Z})$ on $\mathbb{Z}$ such that orbits under each $g \in G$ are finite?
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0answers
127 views

Zariski dense subgroups and conjugates

Let $H \leq \mathrm{SL}_3(\mathbb{Z})$ be a finitely generated subgroup which is not Zariski dense, and let $g \in \mathrm{SL}_3(\mathbb{Z})$. Must there be some $a \in \mathrm{SL}_3(\mathbb{Z})$ such ...
1
vote
0answers
75 views

Are lattices in the special real linear group subgroup seperable?

Let $G \leq SL_2(\mathbb{R})$ be a lattice, let $H \leq G$ be a finitely generated subgroup of infinite index, and let $n \in \mathbb{N}$. Must there be some $H \leq U \leq G$ such that $n \leq [G : ...
2
votes
2answers
207 views

F.g group with infinite ends not Q.I to a free group

Is there any easy example of a finitely generated group with a Cantor set of ends that is not quasi-isometric to a finitely generated free group? Thanks in advance.
1
vote
0answers
94 views

Can a profinite completion be free pro-p?

Is there a prime number $p$ and a finitely generated residually finite group whose profinite completion is a free pro-$p$ group on a nonempty finite set? Thanks to YCor we see that we cannot take the ...
3
votes
1answer
190 views

Are there quasiconvex normal subgroups?

Let $G$ by a hyperbolic group, and let $H \lhd G$ be a normal quasiconvex subgroup. Is it possible that $|H| = [G : H] = \infty$ ?
2
votes
0answers
157 views

Exotic actions of hyperbolic groups

Let $G$ be a hyperbolic group acting faithfully on $\mathbb{Z}$ such that: The action is highly transitive - it is $k$-transitive for each $k \in \mathbb{N}$. For every quasiconvex subgroup $H \leq ...
5
votes
1answer
207 views

A hyperbolic group with a small profinite completion

Is there a finitely generated non-elementary word hyperbolic group the profinite completion of which is known (or conjectured) to be rather restricted, that is: abelian, pro-$p$, virtually ...
6
votes
2answers
297 views

Is residual finiteness a property of “many” finitely presented groups?

Is there a reasonable random model for selecting a finitely presented group $G$ such that with positive probablity (or even with probability almost $1$) some of the following properties hold: $G$ is ...
6
votes
1answer
146 views

Is there a highly transitive action of a finitely generated torsion simple group?

Is there a highly transitive action of a finitely generated torsion simple group $G$ on $\mathbb{Z}$ ? Highly transitive means $k$-transitive for each $k \in \mathbb{N}$, that is: for every two ...
4
votes
1answer
212 views

Characterizing residually amenable groups

Let $G$ be a finitely generated group. The amenability of $G$ is equivalent to the existence of a certain "weak measure" on $G$. Is there such a characterization for residually amenable groups as ...
5
votes
0answers
94 views

Centralizers in virtually special groups

Let $G$ be a virtually compact special group, in the terminology of Haglund and Wise (i.e., $G$ has a finite index subgroup $H$ which is isomorphic to the fundamental group of some compact special ...
3
votes
0answers
147 views

Growth of the number of generators in hyperbolic groups

Let $G$ be an infinite hyperbolic group, and let us further assume that it is residually finite (or even LERF/GFERF) so that we have plenty of subgroups of finite index. I would like to know if one ...
2
votes
1answer
128 views

Measuring products of finitely generated subgroups of free groups

Let $F$ be a finitely generated free group, $H_1, \dots, H_n$ finitely generated subgroups of infinite index in $F$, and $\epsilon > 0$. Must there be an epimorphism to a finite group $\phi \colon ...
8
votes
2answers
292 views

Can a positive measure subset of a free group be nowhere dense?

Let $F$ be a finitely generated free group and let $S \subseteq F$ be a subset for which there is some $\epsilon > 0$ such that for any epimorphism to a finite group $\phi \colon F \to G$ we have ...
1
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0answers
95 views

Free abelian subgroups and distorsion

I realized that I know groups with distorted cyclic subgroups and groups all of whose free abelian subgroups are undistorted, but nothing between. Maybe it is a naive question, but: Does there ...
5
votes
1answer
208 views

Is there a left orderable profinite group?

Is there a profinite group $G$ with a binary transitive relation $<$ such that for any different $x,y \in G$ either $x < y$ or $y < x$ and such that for any $x,y,z \in G$ we have that $x < ...
6
votes
1answer
252 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 ...
1
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0answers
98 views

How to find a invariant surface of a diffeomorphism

Recently, I read a paper about discrete Schrödinger operator. There is a map related to trace map from $C^3$ to $C^3$ as follows: $$T(x,y,z)=(y,z,yz-x).$$ We can calculated that $T$ has the folliwng ...
7
votes
2answers
225 views

Is the free abstract group residually of rank d > 2?

Let $d \geq 2$ be an integer, and let $\mathcal{F}_d$ be the family of finite groups such that $G \in \mathcal{F}_d$ if and only if every subgroup of $G$ can be generated by at most $d$ elements. Is ...
0
votes
1answer
99 views

Example of a polycyclic group which is not of polynomial growth? [closed]

The title already says everything: What is an example of a polycyclic group $G$ which is not of polynomial growth (equivalently, by Gromov's theorem, which is not virtually nilpotent)?
8
votes
1answer
211 views

amenable + without $BS(m,n)$+finite $K(G,1)$implies virtually cyclic?

I heard from someone that the following problem is an open question. (Open Problem 1)For a countable discrete group $G$, suppose it does not contain any Baumslag-Solitar subgroups $BS(m,n):=\langle ...
1
vote
1answer
340 views

Word metrics and finite index subgroups

Suppose that we are given some finitely generated group $ G $ and some finite index subgroup of it $ H $. Given a finite generating symmetric generating set $ S \subset G $, we can define the word ...
2
votes
1answer
111 views

Open Books $( \Sigma, \Phi) $ living in Lefschetz Fibrations over the disk $D^2$

I have a question about open books and Lefschetz fibrations over the 2-disk $D^2$. Please let me set it up first, before going on. Setup: Say we have a Lefschetz fibration $f: W^4 \rightarrow D^2 $ ...
14
votes
3answers
493 views

Failure of Mostow rigidity in dim. 2

I am trying to understand why Mostow rigidity fails in dimension 2. More concretely, I have the following question: (1) What is an example of a quasiisometry $f$ of the hyperbolic plane $\mathbb H^2$ ...
10
votes
1answer
186 views

Cantor-Bernstein for quasi-isometric embeddings?

Suppose that two finitely generated groups quasi-isometrically embed into each other. Does it follow that the two groups are quasi-isometric? Recall that a quasi-isometry is a quasi-isometric ...
5
votes
2answers
256 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 $G$ ...
2
votes
0answers
134 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 $H \leq \Gamma_g$ ...
4
votes
0answers
86 views

Actions of amenable groups on graphs with uncountably many ends

Let $G$ be a finitely generated amenable group acting transitively on an amenable Schreier graph $S$. Is it possible for $S$ to have uncountably many ends? An amenable graph with uncountably many ends ...