arXiv subjects

Collin Bleak

Publications and source records attributed to Collin Bleak.

At least 37 records · Page 2Linked to original sources

arXiv2016

Determining solubility for finitely generated groups of PL homeomorphisms

The set of finitely generated subgroups of the group $PL_+(I)$ of orientation-preserving piecewise-linear homeomorphisms of the unit interval includes many important groups, most notably R.~Thompson's group $F$. In this paper we show that every finitely generated subgroup $G<PL_+(I)$ is either soluble, or contains an embedded copy of Brin's group $B$, a finitely generated, non-soluble group, which verifies a conjecture of the first author from 2009. In the case that $G$ is soluble, we show that the derived length of $G$ is bounded above by the number of breakpoints of any finite set of generators. We specify a set of `computable' subgroups of $PL_+(I)$ (which includes R. Thompson's group $F$) and we give an algorithm which determines in finite time whether or not any given finite subset $X$ of such a computable group generates a soluble group. When the group is soluble, the algorithm also determines the derived length of $\langle X\rangle$. Finally, we give a solution of the membership problem for a family of finitely generated soluble subgroups of any computable subgroup of $PL_+(I)$.

math.GR↗

arXiv2016

An exploration of normalish subgroups of R. Thompson's groups $F$ and $T$

In this short note, we show that R. Thompson's group $F$ admits a normalish amenable subgroup, and that the standard copy of $F$ in R. Thompson's group $T$ is normalish in $T$. We further conjecture that if $F$ is non-amenable, then $T$ does not admit a normalish amenable subgroup, and therefore that the reduced $C^*$ algebra of $T$ is in fact simple in that case.

math.GR↗

arXiv2016

Embedding Right-Angled Artin Groups into Brin-Thompson Groups

We prove that every finitely-generated right-angled Artin group can be embedded into some Brin-Thompson group $nV$. It follows that many other groups can be embedded into some $nV$ (e.g., any finite extension of any of Haglund and Wise's special groups), and that various decision problems involving subgroups of $nV$ are unsolvable.

math.GR↗

arXiv2016

A dynamical definition of f.g. virtually free groups

We show that the class of finitely generated virtually free groups is precisely the class of demonstrable subgroups for R. Thompson's group $V$. The class of demonstrable groups for $V$ consists of all groups which can embed into $V$ with a natural dynamical behaviour in their induced actions on the Cantor space $\mathfrak{C}_2 := \left\{0,1\right\}^ω$. There are also connections with formal language theory, as the class of groups with context-free word problem is also the class of finitely generated virtually free groups, while R. Thompson's group $V$ is a candidate as a universal $co\mathcal{CF}$ group by Lehnert's conjecture, corresponding to the class of groups with context free co-word problem (as introduced by Holt, Rees, Röver, and Thomas). Our main reults answers a question of Berns-Zieze, Fry, Gillings, Hoganson, and Matthews, and separately of Bleak and Salazar-Díaz, and it fits into the larger exploration of the class of $co\mathcal{CF}$ groups as it shows that all four of the known closure properties of the class of $co\mathcal{CF}$ groups hold for the set of finitely generated subgroups of $V.$

math.GR↗

arXiv2014

Some isomorphism results for Thompson like groups $V_n(G)$

We consider a class of groups $V_n(G)$ which are supergroups of the Higman-Thompson groups $V_n$. These groups fit in a framework of Elizabeth Scott for generating infinite virtually simple groups, and the groups we study in particular are initially introduced by Farley and Hughes. The group $V_n(G)$ is the result one obtains by taking the $V_n$ generators and adding a tree automorphism for each generator of a subgroup $G$ of the symmetric group on $n$ letters, where the new generators each permute the child leaves of a specific vertex $α$ of the infinite rooted $n$-ary tree according to the permutation they represent, and then they iterate this permutation again at each vertex which is a descendent of $α$. Farley and Hughes show that $V_n(G)$ is not isomorphic to $V_n$ when $G$ fails to act freely on the points $\{1,2,...,n\}$, and expect further non-isomorphism results in the other cases. We show the perhaps surprising result that if $G$ does act freely, then $V_n(G)\cong V_n$. We also generalise these results and produce some examples of even more isomorphisms amongst groups in the family $V_n(G)$. Essential tools in the above work are a study of the dynamics of the action of elements of $V_n(G)$ on Cantor space, Rubin's Theorem, and transducers from Grigorchuk, Nekrashevych, and Suschanskiĭ's rational group on the $n$-ary alphabet.

math.GR↗

arXiv2014

Ideal structure of the C*-algebra of Thompson group T

In a recent paper Uffe Haagerup and Kristian Knudsen Olesen show that for Richard Thompson's group $T$, if there exists a finite set $H$ which can be decomposed as disjoint union of sets $H_1$ and $H_2$ with $\sum_{g\in H_1}π(g)=\sum_{h\in H_2}π(h)$ and such that the closed ideal generated by $\sum_{g\in H_1}λ(g)-\sum_{h\in H_2}λ(h)$ coincides with $C^*_λ(T)$, then the Richard Thompson group $F$ is not amenable. In particular, if $C_λ^*(T)$ is simple then $F$ is not amenable. Here we prove the converse, namely, if $F$ is not amenable then we can find two sets $H_1$ and $H_2$ with the above properties. The only currently available tool for proving simplicity of group $C^*$-algebra is Power's condition. We show that it fails for $C_λ^*(T)$ and present an apparent weakening of that condition which could potentially be used for various new groups $H$ to show the simplicity of $C_λ^*(H)$. While we use our weakening in the proof of the first result, we also show that the new condition is still too strong to be used to show the simplicity of $C_λ^*(T)$. Along the way, we give a new application of the Ping-Pong Lemma to find free groups as subgroups in groups of homeomorphisms of the circle generated by elements with rational rotation number.

math.OA↗

arXiv2014

Some undecidability results for asynchronous transducers and the Brin-Thompson group 2V

Using a result of Kari and Ollinger, we prove that the torsion problem for elements of the Brin-Thompson group 2V is undecidable. As a result, we show that there does not exist an algorithm to determine whether an element of the rational group R of Grigorchuk, Nekrashevich, and Sushchanskii has finite order. A modification of the construction gives other undecidability results about the dynamics of the action of elements of 2V on Cantor Space. Arzhantseva, Lafont, and Minasyanin prove in 2012 that there exists a finitely presented group with solvable word problem and unsolvable torsion problem. To our knowledge, 2V furnishes the first concrete example of such a group, and gives an example of a direct undecidability result in the extended family of R. Thompson type groups.

math.GR↗

arXiv2013

Embeddings into Thompson's group $V$ and $co\mathcal{CF}$ groups

Lehnert and Schweitzer show in [20] that R. Thompson's group $V$ is a co-context-free ($co\mathcal{CF}$) group, thus implying that all of its finitely generated subgroups are also $co\mathcal{CF}$ groups. Also, Lehnert shows in his thesis that $V$ embeds inside the $co\mathcal{CF}$ group $\mathrm{QAut}(\mathcal{T}_{2,c})$, which is a group of particular bijections on the vertices of an infinite binary $2$-edge-colored tree, and he conjectures that $\mathrm{QAut}(\mathcal{T}_{2,c})$ is a universal $co\mathcal{CF}$ group. We show that $\mathrm{QAut}(\mathcal{T}_{2,c})$ embeds into $V$, and thus obtain a new form for Lehnert's conjecture. Following up on these ideas, we begin work to build a representation theory into R. Thompson's group $V$. In particular we classify precisely which Baumslag-Solitar groups embed into $V$.

math.GR↗

arXiv2011

Centralizers in R. Thompson's group V_n

Let n be bigger than 1 and let A be an element in the Higman-Thompson group V_n. We study the structure of the centralizer of a in V_n through a careful analysis of the action of the group generated by A on the Cantor set C. We make use of revealing tree pairs as developed by Brin and Salazar from which we derive discrete train tracks to assist us in our analysis. A consequence of our structure theorem is that centralizers are finitely generated. Along the way we give a short argument using revealing tree pairs which shows that cyclic groups are undistorted in V_n.

math.GR↗

arXiv2011

Structure theorems for subgroups of homeomorphisms groups

In this partly expository paper, we study the set A of groups of orientation-preserving homeomorphisms of the circle S^1 which do not admit non-abelian free subgroups. We use classical results about homeomorphisms of the circle and elementary dynamical methods to derive various new and old results about the groups in A. Of the known results, we include some results from a family of results of Beklaryan and Malyutin, and we also give a new proof of a theorem of Margulis. Our primary new results include a detailed classification of the solvable subgroups of R. Thompson's group T .

math.GR↗

arXiv2009

Free products in R. Thompson's group V

We investigate free product structures in R. Thompson's group V, primarily by studying the topological dynamics associated with V's action on the Cantor Set. We show that the class of free products which can be embedded into V includes the free product of any two finite groups, the free product of any finite group with Q/Z, and the countable non-abelian free groups. We also show the somewhat surprising result that Z^2*Z does not embed in V, even though V has many embedded copies of Z^2 and has many embedded copies of free products of pairs of its subgroups.

math.GR↗

arXiv2008

A family of non-isomorphism results

We give a short argument showing that if $m, n \in {1, 2, ...} \cup ω$, then the groups mV and nV are not isomorphic. This answers a question of Brin.

math.GR↗

arXiv2007

Finite index subgroups of R. Thompson's group F

The authors classify the finite index subgroups of R. Thompson's group $F$. All such groups that are not isomorphic to $F$ are non-split extensions of finite cyclic groups by $F$. The classification describes precisely which finite index subgroups of $F$ are isomorphic to $F$, and also separates the isomorphism classes of the finite index subgroups of $F$ which are not isomorphic to $F$ from each other; characterizing the structure of the extensions using properties of the structure of the finite index subgroups of $Z\times Z$.

math.GR↗

arXiv2007

Twisted conjugacy classes in R. Thompson's group F

In this short article, we prove that any automorphism of the R. Thompson's group $F$ has infinitely many twisted conjugacy classes. The result follows from the work of Matthew Brin, together with a standard facts on R. Thompson's group $F$, and elementary properties of the Reidemeister numbers.

math.GR↗

arXiv2006

Some questions about the dimension of a group action

We discuss three families of groups, $ZW_n$, $PL(I^n)$, and $PL(S^n)$ (the last two being families of groups of piecewise-linear homeomorphisms of standard $n$-dimensional spaces). We note that for all positive integers $n$, $ZW_n$ embeds in $PL(I^n)$ which embeds in $PL(S^n)$. In another direction, $ZW_2$ fails to embed in $PL(I^1)$ by an earlier result of the author, and we extend that result to show that $ZW_2$ also fails to embed in $PL(S^1)$. The nature of the proofs of these non-embedding results leads us to ask if there are corresponding non-embedding results in higher dimensions.

math.GR↗

arXiv2006

A Minimal Non-solvable Group of Homeomorphisms

Let $PL_0(I)$ represent the group of orientation-preserving piecewise-linear homeomorphisms of the unit interval which admit finitely many breaks in slope, under the operation of composition. We find a non-solvable group $W$ and show that $W$ embeds in every non-solvable subgroup of $PL_0(I)$. We find mild conditions under which other non-solvable subgroups ($B$, $(\wr\Z\wr)^{\infty}$, $(\Z\wr)^{\infty}$, and $^{\infty}(\wr\Z)$) embed in subgroups of $PL_0(I)$. We show that all solvable subgroups of $PL_0(I)$ embed in all non-solvable subgroups of $PL_0(I)$. These results continue to apply if we replace $PL_0(I)$ by any generalized R. Thompson group $F_n$.

math.GR↗

arXiv2006

An algebraic classification of some solvable groups of homeomorphisms

We produce two separate algebraic descriptions of the isomorphism classes of the solvable subgroups of the group PLo(I) of piecewise-linear orientation-preserving homeomorphisms of the unit interval under the operation of composition, and also of the solvable subgroups of the generalized R. Thompson groups F_n. The first description is as a set of isomorphism classes of groups which is closed under three algebraic operations, and the second is as the set of isomorphism classes of subgroups of a countable collection of easily described groups. We show the two descriptions are equivalent.

math.GR↗

arXiv2006

A geometric classification of some solvable groups of homeomorphisms

We investigate subgroups of the group PLo(I) of piecewise-linear, orientation-preserving homeomorphisms of the unit interval with finitely many breaks in slope, under the operation of composition, and also subgroups of the generalized Thompson groups F_n. We find geometric criteria determining the derived length of any such group, and use this criteria to produce a geometric classification of the solvable and non-solvable subgroups of PLo(I) and of the F_n. We also show that any standard restricted wreath product C wr T (of non-trivial groups) that embeds in PLo(I) or F_n must have T isomorphic with the integers.

math.GR↗