# Questions tagged [geometric-group-theory]

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

822
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### Density of ``diagonal sets'' in amenable groups

Let $G$ be a countable amenable group with a (left) Følner sequence $(F_n)$. Let $\Gamma$ be a subset of $G$ that has density $1$ with respect to $(F_n)$, in the sense that
$$
\lim_{n \to \infty} \...

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### Does there exist a finitely generated, torsion group $G$ with a residually finite ascending HNN extension?

I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can provide me some insight.
Let $G$ be a group with an injective endomorphism $\phi$...

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1
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### When are groups generated by reflections in a triangle discrete?

Take a triangle in the (Euclidean or hyperbolic) plane, and consider the group of isometries generated by the reflections in the three sides of the triangle. If the angles between adjacent sides are ...

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### When is a generalised Baumslag-Solitar group linear?

$\DeclareMathOperator\BS{BS}$The linearity of the Baumslag-Solitar groups $\BS(m, n)=\langle a, t\mid t^{-1}a^mt=a^n\rangle$ is completely understood, and it may be phrased as: $\BS(m, n)$ is linear ...

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### Let $H$ be a co-dimension 1 quasiconvex subgroup of a one-ended hyperbolic group $H$. Does $H$ permute the components of $\partial G - \Lambda H?$

Let $H$ be a co-dimension 1 quasiconvex subgroup of a one-ended hyperbolic group $G$. In particular, $\partial G$ is connected but $\partial G - \Lambda H$ is disconnected. The number of components of ...

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### Hilbert space compression of CAT(0) groups

Does there exist a CAT(0) group with Hilbert space compression $<1$?
The Hilbert space compression of a metric space $(X,d)$, e.g. a group endowed with the word metric given by a finite generating ...

3
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1
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148
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### Residual finiteness of random groups with property (T)

A well known result of A. Zuk states that for $\frac{1}{3} < d < \frac{1}{2}$, a random group $\Gamma$ with respect to Gromov's density model with density $d$ has Kazhdan's property (T) with ...

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### Existence of properly discontinuous and cocompact action

Let $M$ be a complete Riemannian manifold. My question is, under what conditions does $M$ admit a discrete group of isometries $\Gamma$ which acts properly discontinuously and cocompactly on $M$, that ...

8
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3
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### Proving that a countable group is not finitely generated

I would like to learn about techniques for proving that a countable group is not finitely generated. I am also interested in learning about examples. Finally, I am particularly, but not exclusively, ...

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2
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### HNN decomposition of finite rank free group over infinite rank subgroups

It's a nice result of Swarup that whenever a free group $G$ splits as an HNN extension $G = J \ast_{H,t}$ with $H$ a finitely generated subgroup, there exist splittings $J = J_1 \ast J_2$ and $H = H_1 ...

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### Examples of hyperbolic groups with non-hyperbolic subgroups

In a previous question, I asked about hyperbolic groups in which every finitely generated subgroup is hyperbolic. I am now curious about the reverse question: what are some examples of hyperbolic ...

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### Examples of locally hyperbolic groups

It is well-known that a subgroup of a hyperbolic group need not be hyperbolic. Let us say that a (finitely generated) group $G$ is locally hyperbolic if all its finitely generated subgroups are (...

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1
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### Examples of group families with solvable uniform word problem

I would like to know of any examples of families of groups that are known (or conjectured) to have a solvable uniform word problem, i.e. an algorithm that given a presentation $P$ of a group in the ...

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### Type $F_n$ and quasi-isometry with highly connected space

Recall that a group is of type $F_n$ if it has a classifying space with finite $n$-skeleton. For example type $F_1$ means finitely generated, and type $F_2$ means finitely presented.
Question: For a ...

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1
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### Looking for a citation: the Rips $n$-complex of a $\delta$-hyperbolic group is contractible for high enough $n$

Given a $\delta$-hyperbolic group $G$, I have been told that the Rips $n$-complex of $G$ is contractible for high enough $n$. The only proof I have found for this statement is in an expository essay ...

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### Example of CAT($k$) space [closed]

Good time of day. I repeat the question from MSE (https://math.stackexchange.com/questions/4464888/question-about-example-of-catk-space) because no response has been received.Question is the following:...

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### Question about coarse fixed point property in large-scale geometry

I read the article of Steven Hair "A degree-theoretic proof of a coarse fixed point principle". I have the following question.
I start with some main definitions from this article. A coarse ...

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490
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### Mapping Out(F_n) to the mapping class group

Let $\mathrm{Out}(F_g)$ denote the automorphism group of a free group, and $\mathrm{Mod}_g$ the mapping class group of a closed oriented genus $g$ surface. Is there a map, as indicated with the dashed ...

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### Reduced group C*-algebra $C^*_r(\mathbb{Z}/2*\mathbb{Z}/2)$: norm of specific elements

Consider the free product of $\mathbb{Z}/2$ with itself with generators
$$
\mathbb{Z}/2*\mathbb{Z}/2=\langle u,v\mid u^2=1=v^2\rangle
$$
and regard its group $C^*$-algebra
$$
C^*(\mathbb{Z}/2*\mathbb{...

3
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1
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### Carne-Varopoulos bound and stationary measure

Let $\Gamma$ denote the Cayley graph for a finitely generated group $G$, and let $p_n(x, y)$ denote the transition probability that a random walk starting at $x$ reaches $y$ at time $n$. A famous &...

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### The figure eight knot complement in $S^3$

Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. Exercise 5.4 (on page 101) gives us a presentation of the fundamental group of $S^3 - K$ where $K$ is the figure-...

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1
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### Finiteness of $\ell^2$-Betti numbers

I'm reading the paper "Improved algebraic fibering" by Sam Fisher (https://arxiv.org/pdf/2112.00397.pdf) and in the proof of lemma 6.4 it claims the followng:
$(\mathcal{D}_{\mathbb{F}K}\ast\...

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### Distortion in the Brin-Thompson 2V

Is it known whether the Brin-Thompson 2V contains a distortion element? By this I mean an element $f$ such that the word norm $|f^n|$
grows sublinearly, and $f$ is of infinite order. If such an ...

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1
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### Are the non-free factors of Grushko decomposition of a finitely generated convex-cocompact (but not cocompact) subgroup of PSL$(2,\mathbb{R})$ finite?

See Grushko decomposition theorem.
Are the non-free factors of Grushko decomposition of a finitely generated convex–cocompact (but not cocompact) subgroup of $\operatorname{PSL}(2,\mathbb{R})$ finite?
...

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### What about a Cayley n-complex for n>2?

Let $G$ be a finitely presented group. The Cayley graph of the finite generating set is a $1$-complex where the $0$-cells are the elements of $G$ and the $1$-cells are given by the generators (...

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### Inverse limit in category of graphs

Let $... \leftarrow G_i \leftarrow G_{i+1} \leftarrow ...$ be an inverse system of graphs (which might not be locally finite), morphisms are symplicial maps (we allow collapsing of edges to vertices)....

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### Convex core and geometric finiteness of negatively curved manifolds

I am reading a paper on hyperbolic geometry where they are using the concept of "convex core" in the context of "geometric finiteness". Roughly, this means (from Definition F4 of a ...

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2
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### Group of exponential growth always contains a free sub-group?

I am not very conversant with the growth of a group, so this may be a very silly question.
It is known that $F_2$, the free group of rank $2$, has exponential growth. I was wondering whether the ...

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0
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### Ends of a negatively curved Riemannian manifold

Let $M$ be a complete Riemannian manifold. Let us use the standard definition of "end", for example, as in this article. If $M$ has non-negative Ricci curvature, it is well-known that it has ...

7
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4
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391
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### Is every virtually free group residually finite?

Question: Is every (finitely generated) virtually free group residually finite?
A well-known question asks whether every hyperbolic group is residually finite (Mladen Bestvina. Questions in geometric ...

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### Decomposition about splitting of symmetric spaces of compact type

I get stuck in the following question:
Why does a locally symmetric space of compact type $M$ split locally irreducible components of dimension $\geq 2$ which are Einstein? In particular, why are all ...

5
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1
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### Is this semi-direct product residually finite?

I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can help me find a way to check the residual finiteness of this group.
Consider the ...

29
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1
answer

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### Group theory with grep?

While reading Bill Thurston's obituary in the Notices of the AMS I came across the following fascinating anecdote (pg. 32):
Bill’s enthusiasm during the early stages of mathematical discovery was ...

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1
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89
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### Number of orbits for abelian group actions

Suppose $G$ is an abelian group acting faithfully on two sets, $X$ and $Y$, of the same size. None of $G$, $X$ and $Y$ is finite.
Now suppose $G$ is the union of abelian groups $G_i$, where $i$ varies ...

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1
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### Parahoric subgroup over a local field

$\DeclareMathOperator\SL{SL}$Let $F$ be a local field and $\mathcal{O}_{F}$ its valuation ring. Let $\pi\in \mathcal{O}_{F}$ be a uniformizer and $\mathfrak{p}=\pi\mathcal{O}_{F}$. Let $G$ be a split ...

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### Counterexamples to an analog of Cannon's Conjecture which do not arise from manifolds?

Let $\Gamma$ be a finitely presented hyperbolic group with boundary homeomorphic to $S^{n-1}$. Are there any examples of such $\Gamma$ which are known to not be the fundamental group of any $n$-...

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1
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### Permuting subgroups with the same finite index

Suppose that we have a finitely generated residually finite group $G = \langle g_1,\ldots,g_r \rangle$ with polynomial growth. Let $H$ be a subgroup of $G$ with finite index $m$. Let $\phi$ be an ...

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### Positive harmonic functions on nilpotent groups & Random walk on groups with a finite number of generators

I want to read the following papers in the English version which I could not find anywhere (the only papers I can get are the Russian versions). Kindly help me out.
Gregory A. Margulis, Positive ...

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3
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387
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### Spectral radius of a finitely generated group

Let $G$ be a finitely generated group and $\Gamma$ be its Cayley graph with the usual word metric. Let $\mu$ be a symmetric non-degenerate measure on $G$ (maybe with finite support or smooth), and ...

5
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### If $K\rtimes \mathbb{Z}$ is a finitely generated group but $K$ isn't, must the fixed points of $1_\mathbb{Z}$ be a finitely generated group?

I've copied over this question from what I asked on StackExchange, in the hope that an expert here can readily answer the question.
Is there an example of a group $G=K\rtimes \mathbb{Z}$ satisfying ...

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### Discrete cocompact group of isometries of Nil

Is it true that every group quasi-isometric to the Heisenberg group admits a proper cocompact action by isometries on Nil?

5
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1
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### Given a quasi-convex subgroup $H$ of hyperbolic $G$, can we decide if two elements $x,y \in G$ lie in the same double coset of $H$?

I've come across the following question in my research, which seems elusive but is almost surely decidable.
Let $H$ be a quasi-convex subgroup of the hyperbolic group $G$. Given $x, y \in G$, we wish ...

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### Infinite groups that admit a discrete, co-compact, bilipschitz action on $\mathbb{R}^3$

Apart from the abstract types of the crystallographic groups, are there any other abstract groups that admit a proper, co-compact, uniformly bilipschitz action on $\mathbb{R}^3$?

3
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### First Betti number of finitely non-co-hopfian groups

Let $G$ be a finitely generated group. Assume that $G$ is a finitely non-co-hopfian group in the sense that there is a group embedding $i: G\hookrightarrow G$ such that $1<[G\colon i(G)]<\infty$...

5
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2
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### Potential counterexamples to Bass' trace conjecture

Motivation: The following is a theorem of Berrick-Hesselholt (essentially also due to Linnell, though not in this form):
Let $G$ be a group. Suppose that for every subgroup of $G$ isomorphic to $\...

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1
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90
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### Are geometric actions on CAT(0) spaces with isolated flats minimal on the boundary?

Suppose $X$ is a $CAT(0)$ space with isolated flats, $\partial X$ its visual boundary and $G$ acts properly discontinuously amd cocompactly on $X$. Must the $G$ action on $\partial X$ have a dense ...

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1
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### Torsion-free subgroup of affine group

Let $G$ be a finitely generated group and $\varphi:G\to \operatorname{Aut}(\mathbb C)$ a homomorphism, where $\operatorname{Aut}(\mathbb C)$ is the group of complex affine transfromations $a z+b$.
Can ...

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0
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### Finitely presentable group with purely infinite full group $C^*$-algebra?

Does there exist an example of a finitely presentable group whose full group $C^*$-algebra is purely infinite,
resp. is it known to be impossible?

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### Non-finitely presented FP groups with cohomological dimension $2$

In this recent preprint, the authors construct a certain uncountable family of non-finitely presented FP groups. Recall that group is an FP group if the trivial $\mathbb Z[G]$-module $\mathbb Z$ has a ...

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### Which properties can be read off the balls of a Cayley graph?

For which properties (P) [of groups] does the following hold:
given a group $G$ which has a finite presentation with at most $n$ relations of length at most $\ell$, there is a $R(n,\ell)$ so that, if ...