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Results tagged with geometric-group-theory
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user 28104
Large scale properties of groups; growth functions; Dehn functions; small cancellation properties; hyperbolicity and CAT(0); actions and representations; combinatorial group theory; presentations
6
votes
Group of exponential growth always contains a free sub-group?
Not necessarily. -- For example, the lamplighter group has exponential growth, but does not have a free subgroup of rank 2 (if $a$ and $b$ are two elements of that group of infinite order, then there …
- 19.6k
10
votes
Accepted
Is it true that every f.g. infinite simple group has exponential growth?
No, there exists a finitely generated infinite simple group of intermediate
growth. This has meanwhile been found out by Volodymyr Nekrashevych, cf.
Palindromic subshifts and simple periodic groups o …
- 19.6k
3
votes
Is there a highly transitive action of a finitely generated torsion simple group?
As you expressed interest in Thompson's group V in the comments --
the following is a highly transitive faithful permutation representation of V on $\mathbb{Z}$ (although as Ives de Cornulier has alre …
- 19.6k
16
votes
1
answer
913
views
Is it true that every f.g. infinite simple group has exponential growth?
Is it true that every finitely generated infinite simple group has
exponential (word-)growth?
Remark: As Mark Sapir has pointed out, the question whether
every finitely generated group of subexponent …
- 19.6k
3
votes
Proving that a countable group is not finitely generated
One possible way of proving that a countable group $G$ is not finitely
generated is finding an infinite set $S$ and a mapping $\varphi$ from $G$
to the power set of $S$ such that the following hold:
…
- 19.6k