arXiv subjects
Publications and source records attributed to Yiftach Barnea.
We study the virtually generating graph of pro-$p$ groups. We show that various classes of pro-$p$ groups have connected virtually generating graph and we bound its diameter in these cases; e.g.\ compact subgroups of analytic groups over local fields and the Nottingham group.
We study the commensurators of free groups and free pro-$p$ groups, as well as certain subgroups of these. We prove that the commensurator $Comm(F)$ of a non-abelian free group of finite rank $F$ is not virtually simple, answering a question of Lubotzky. On the other hand, we exhibit a family of easy-to-define finitely generated subgroups of $Comm(F)$ and show that some groups in this family are simple. For a prime $p$, we also consider the p-commensurator $Comm_p(F)$, which is the commensurator of $F$ viewed as a group with pro-$p$ topology. By contrast with $Comm(F)$, we prove that $Comm_p(F)$ has a simple subgroup of index at most 2. Further, while the isomorphism class of $Comm(F)$ does not depend on the rank of $F$, we prove that the isomorphism class of $Comm_p(F)$ depends on the rank of $F$ and determine the exact dependency. If $\mathbf F$ is the pro-$p$ completion of $F$ (which is a free pro-$p$ group), $Comm(\mathbf F)$ is a totally disconnected locally compact (tdlc) group containing $\mathbf F$ as an open subgroup. We use $Comm_p(F)$ to construct an abstractly simple subgroup of $Comm(\mathbf F)$ containing $\mathbf F$ as well as a family of non-discrete tdlc groups which are compactly generated and simple.
Given a discrete (resp. profinite) group $G$, we define $NCC(G)$ to be the smallest number of cyclic (resp. procyclic) subgroups of $G$ whose conjugates cover $G$. In this paper we determine all residually finite discrete groups with finite NCC and give an almost complete characterization of profinite groups with finite NCC.
Put $R=\F[[t_1, \ldots, t_d]])$. We estimate the number of normal subgroups of $\mathrm{SL}_2^1(\F[[t_1, \ldots, t_d]])$ for $p>2$, the number of ideals in the Lie algebra $\Lie(R)$, and the number of ideals in the associative algebra $R$.
The maximal normal subgroup growth type of a finitely generated group is $n^{\log n}$. Very little is known about groups with this type of growth. In particular, the following is a long standing problem: Let $Γ$ be a group and $Δ$ a subgroup of finite index. Suppose $Δ$ has normal subgroup growth of type $n^{\log n}$, does $Γ$ has normal subgroup growth of type $n^{\log n}$? We give a positive answer in some cases, generalizing a result of Müller and the second author and a result of Gerdau. For instance, suppose $G$ is a profinite group and $H$ an open subgroup of $G$. We show that if $H$ is a generalized Golod-Shafarevich group, then $G$ has normal subgroup growth of type of $n^{\log n}$. We also use our methods to show that one can find a group with characteristic subgroup growth of type $n^{\log n}$.
We first show that a class of pro-$p$ branch groups including the Grigorchuk group and the Gupta-Sidki groups all have subgroup growth type $n^{\log n}$. We then introduce the notion of orbit growth and use it to construct extensions of the Grigorchuk group and the Gupta-Sidki groups. We compute the subgroup growth type of these extensions and deduce that all functions between $n^{(\log n)^2}$ and $e^n$ occur as the subgroup growth type of a pro-$p$ group, thus, giving almost complete answer to a question raised by Lubotzky and Segal.
We prove that the inverse limit of certain iterated wreath products in product action have complete Hausdorff dimension spectrum with respect to their unique maximal filtration of open normal subgroups. Moreover we can produce explicitly subgroups with a specified Hausdorff dimension.
Recently, Schlage-Puchta proved super multiplicity of $p$-deficiency for normal subgroups of $p$-power index. We extend this result to all normal subgroups of finite index. We then use the methods of the proof to show that some groups with non-positive $p$-deficiency have virtually positive $p$-deficiency. We also compute the $p$-deficiency in some cases such as Fuchsian groups and study related invariants: the lower and upper absolute $p$-homology gradients and the $p$-Euler characteristic.
In this paper we initiate a systematic study of the abstract commensurators of profinite groups. The abstract commensurator of a profinite group $G$ is a group $Comm(G)$ which depends only on the commensurability class of $G$. We study various properties of $Comm(G)$; in particular, we find two natural ways to turn it into a topological group. We also use $Comm(G)$ to study topological groups which contain $G$ as an open subgroup (all such groups are totally disconnected and locally compact). For instance, we construct a topologically simple group which contains the pro-2 completion of the Grigorchuk group as an open subgroup. On the other hand, we show that some profinite groups cannot be embedded as open subgroups of compactly generated topologically simple groups. Several celebrated rigidity theorems, like Pink's analogue of Mostow's strong rigidity theorem for simple algebraic groups defined over local fields and the Neukirch-Uchida theorem, can be reformulated as structure theorems for the commensurators of certain profinite groups.