All Questions
Tagged with geometric-group-theory ra.rings-and-algebras
18 questions
1
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0
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89
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The base group of a wreath product of an abelian group by $ {\mathbb{Z}}$ is a characterstic subgroup
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can direct me to some relevant results.
Let $A$ be a finitely generated abelian group,...
- 823
3
votes
2
answers
468
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How fast does the number of "fixed" points grow compared to the size of the ball in the following group?
I have copied this question from Math.StackExchange, in the hope that some experts here can provide some relevant insight.
Let $M = \oplus_{i\in \mathbb Z} V^{(i)}$ where each $ V^{(i)} \cong \mathbb ...
- 823
4
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0
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453
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Problem 1.8 from Kirby's list
Context
I looked through a book called "Problems in Low-Dimensional Topology", where Rob Kirby lists a set of problems. He provides a list of problems, states their conjectures, and ...
2
votes
0
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60
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upper bound for the exponential conjugacy growth rate for non-virtually nilpotent polycyclic groups
Given $n ≥ 0$, the conjugacy growth function $c(n)$ of a finitely generated group $G$, with respect to some finite generating set $S$, counts the number of conjugacy classes intersecting the ball of ...
- 823
2
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0
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156
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The growth rate of the group $\mathbb{Z}[1/2] \rtimes _\phi \mathbb{Z}$, where $\phi (1)$ corresponds to multiplying every number by $2$
Consider the group $G = \mathbb{Z}[1/2] \rtimes _\phi \mathbb{Z}$, where $\mathbb{Z}[1/2] = \{j/2^m \mid j \in \mathbb{Z}, m\in\mathbb{N} \}$, the dyadic rationals, and for every $n\in \mathbb{Z}$, $...
- 823
2
votes
1
answer
244
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Markov property for groups?
My question again refers to the following article:
Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–1470, doi:10.1007/s00222-023-01200-w, arXiv:...
- 309
8
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2
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645
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Analogous results in geometric group theory and Riemannian geometry?
As you can see from my other question I concern mmyself with the following article at the moment:
Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–...
- 309
3
votes
1
answer
353
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Question to limit groups (over free groups)
My question refers to the following article (to page 26: proof of Theorem 4.1):
Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–1470, doi:10....
- 309
5
votes
1
answer
298
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Does every sequence of group epimorphisms (between finitely generated groups) contain a stable subsequence?
I have a question that is related to the topic of limit groups:
Let $G$ and $H$ be finitely generated groups and let $(\varphi_n: G \to H)_{n \in \mathbb{N}}$ be a sequence of group epimorphisms. Does ...
- 309
16
votes
1
answer
850
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A "simpler" description of the automorphism group of the lamplighter group
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts can point me to some relevant references.
The lamplighter group is defined by the ...
- 823
3
votes
0
answers
132
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the growth rate of poly-$\mathbb{Z}$ group
I am interested in the growth rate of the poly-$\mathbb{Z}$ group. Let $G$ be a poly-$\mathbb{Z}$ group, i.e $$G =(\dots((\mathbb{Z} \rtimes_{\phi_1} \mathbb{Z})\rtimes_{\phi_2} \mathbb{Z}) \rtimes_{\...
- 823
4
votes
0
answers
214
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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 $\phi$...
- 823
18
votes
1
answer
783
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Are there any "simple" monoids with intermediate growth?
The discovery of the Grigorchuk group which has intermediate growth caused a number of other such groups to be found, but they are all fairly complicated, and as far as I know none of them are ...
- 1,947
1
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0
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156
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Does $AA^{-1}$ have the unique product property?
Let $A$ be a finite subset of a torsion free group $G$ with $|A|\geq3$. Does the set $AA^{-1}$ have the "unique product" property (i.e. there exist an element $c\in AA^{-1}$ that is uniquely written ...
- 2,118
5
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0
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166
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Chains of right annihilators in group rings
See the update below
This problem emanates from a question on not-so-simple random walks on finitely generated groups. But to explain the connections would require an extremely long essay.
Let $G$ be ...
- 4,679
7
votes
1
answer
318
views
Is the following module over a group ring necessarily infinitely generated?
Suppose $\Gamma$ is a (finitely presented, but this is probably irrelevant) group, and $M$ is a finitely generated (EDIT: finitely presented) module over $\mathbb{Q}\Gamma$ which is infinite-...
- 934
3
votes
1
answer
393
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Kaplansky Idempotent conjecture and Extension theory
We consider the Idempotent Kaplansky conjecture with $\mathbb{C}$- coefficients, that is the problem of nontrivial idempotents for group algebra $\mathbb{C}\Gamma$ where $\Gamma$ is a torsion free ...
- 356
2
votes
2
answers
757
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Abstract Commensurator Group of $\mathbb{Z}^n$ $Comm(\mathbb{Z}^n)\cong GL(n,\mathbb{Q})$?
Hello! In a paper I read that $\mathrm{Comm}(\mathbb{Z}^n)\cong \mathrm{GL}(n,\mathbb{Q})$. Why is that true? How can I find an isomorphism of this groups?
I know that $\mathrm{Aut}(\mathbb{Z}^n)\...
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