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Gili Golan

Publications and source records attributed to Gili Golan.

At least 19 recordsLinked to original sources
arXiv2026

Every copy of Thompson's group $F$ in $F$ is undistorted

Thompson's group $F$ is the group of all piecewise-linear homeomorphisms of the unit interval whose breakpoints are dyadic and whose slopes are integer powers of $2$. If $H$ is a finitely generated subgroup of a finitely generated group $G$, its distortion measures the difference between the intrinsic word metric of $H$ and the metric induced from $G$. The subgroup is undistorted when these metrics are equivalent. Guba and Sapir asked whether $F$ contains a distorted copy of itself. The same question was also suggested by Brin. We answer this question negatively: every subgroup of $F$ isomorphic to $F$ is undistorted in $F$.

math.GR↗
arXiv2026

Irreducible fast sets of bump homeomorphisms generate copies of Thompson's groups $F_n$

A homeomorphism of an interval is a positive bump if its support is a single open interval on which it moves every point to the right. Choosing a fundamental domain $[m,b(m))$ for the action of a positive bump $b$ on its support splits the remainder of the support into two intervals, called the feet of $b$. A finite set of positive bumps is geometrically fast if fundamental domains can be chosen so that all the resulting feet are pairwise disjoint. The crossing graph of such a set has the bumps as its vertices, two bumps being adjacent whenever their supports overlap but are not nested, and the set is irreducible if its crossing graph is connected. We prove that for every $n\geq 2$, every group generated by an irreducible geometrically fast set of $n$ positive bumps is isomorphic to the $n$-ary Thompson group $F_n$. This answers the strong version of a problem posed by Brin and Zaremsky (Oberwolfach Rep. 15 (2018)).

math.GR↗
arXiv2026

Higman--Thompson groups $F_n$ all the way down

We prove that for every $n\ge 2$ the Higman--Thompson group $F_n$ has a maximal subgroup of infinite index isomorphic to itself. In fact, we construct a chain of subgroups $F_n=H_0>H_1>H_2>\cdots$, all isomorphic to $F_n$ and with trivial intersection, such that for every $i$ the only subgroups of $F_n$ containing $H_i$ are $H_i,H_{i-1},\ldots,H_0=F_n$; in particular, each $H_{i+1}$ is maximal in $H_i$. We prove that for all $n\ge m\ge 2$, every closed maximal subgroup of $F_m$ isomorphic to $F_n$ arises from a homeomorphism between the $n$-ary and $m$-ary Cantor spaces given by a finite semi-synchronizing transducer--a variation of the synchronizing transducers of Bleak, Cameron, Maissel, Navas and Olukoya. We characterize the homeomorphisms of Cantor spaces conjugating $F_n$ into $F_m$ as the order-preserving or order-reversing rational homeomorphisms whose minimal transducer is semi-synchronizing. At the heart of the paper is a machinery bridging transducers and Stallings $2$-cores of subgroups, which reduces the conjugation of finitely generated closed subgroups by such homeomorphisms to an algorithmic procedure. As applications, we prove that Jones' ternary oriented subgroup $\vec F_3\le F_3$ is isomorphic to $F_4$, answering questions of Aiello, and that all known maximal subgroups of infinite index of Thompson's group $F$ which act minimally on $(0,1)$ are isomorphic to Higman--Thompson groups. That raises the problem of whether all maximal subgroups of infinite index of $F$ which act minimally on $(0,1)$ are isomorphic to Higman--Thompson groups. We briefly discuss related results regarding fast groups of homeomorphisms and maximal subgroups of Thompson groups.

math.GR↗
arXiv2026

On the generation problem in Thompson's groups $F_n$

We study the generation problem in the Higman-Thompson groups $F_n$ via the core and closure of subgroups of $F_n$ and associated automata. We give sufficient conditions for a subset $X \subseteq F_n$ to generate $F_n$, and provide an algorithm which verifies these conditions when $X$ is finite. As an application, we answer a question of Aiello and Nagnibeda, motivated by Savchuk's problem on maximal subgroups of Thompson's group $F$. Specifically, we show that for every $n\geq 2$, the Higman-Thompson group $F_n$ contains a maximal subgroup of infinite index which fixes no point of $(0,1)$. The subgroup we construct is isomorphic to $F_{2n-1}$.

math.GR↗
arXiv2024

Remembering Mark Sapir

This memorial article for Mark Sapir provides a brief overview of his life and career. Among his many contributions we highlight two of his most celebrated achievements: his groundbreaking solutions to Burnside-type problems for semigroups and his innovative construction of S-machines. Additionally, reflections from his colleagues and friends offer a heartfelt tribute, blending professional insights with personal memories.

math.HO↗
arXiv2024

The "spread" of Thompson's group $F$

Recall that a group $G$ is said to be $\frac{3}{2}$-generated if every non-trivial element $g\in G$ has a co-generator in $G$ (i.e., an element which together with $g$ generates $G$). Thompson's group $V$ was proved to be $\frac{3}{2}$-generated by Donoven and Harper in 2019. It was the first example of an infinite finitely presented non-cyclic $\frac{3}{2}$-generated group. In 2022, Bleak, Harper and Skipper proved that Thompson's group $T$ is also $\frac{3}{2}$-generated. Since the abelianization of Thompson's group $F$ is $\mathbb{Z}$, it cannot be $\frac{3}{2}$-generated. However, we recently proved that Thompson's group $F$ is "almost" $\frac{3}{2}$-generated in the sense that every element of $F$ whose image in the abelianization forms part of a generating pair of $\mathbb{Z}^2$ is part of a generating pair of $F$. A natural generalization of $\frac{3}{2}$-generation is the notion of spread. Recall that the spread of a group $G$ is the supremum over all integers $k$ such that every $k$ non-trivial elements of $G$ have a common co-generator in $G$. The uniform spread of a group $G$ is the supremum over all integers $k$ for which there exists a conjugacy class $C\subseteq G$ such that every $k$ non-trivial elements of $G$ have a common co-generator which belongs to $C$. In this paper we study modified versions of these notions for Thompson's group $F$.

math.GR↗
arXiv2023

On Maximal Subgroups of Thompson's Group $F$

We study subgroups of Thompson's group $F$ by means of an automaton associated with them. We prove that every maximal subgroup of $F$ of infinite index is closed, that is, it coincides with the subgroup of $F$ accepted by the automaton associated with it. It follows that every finitely generated maximal subgroup of $F$ is undistorted in $F$. We also prove that every finitely generated subgroup of $F$ is contained in a finitely generated maximal subgroup of $F$ and construct an infinite family of non-isomorphic maximal subgroups of infinite index in $F$.

math.GR↗
arXiv2022

Thompson's group $F$ is almost $\frac{3}{2}$-generated

Recall that a group $G$ is said to be $\frac{3}{2}$-generated if every non-trivial element of $G$ belongs to a generating pair of $G$. Thompson's group $V$ was proved to be $\frac{3}{2}$-generated by Donoven and Harper in 2019. It was the first example of an infinite finitely presented non-cyclic $\frac{3}{2}$-generated group. Recently, Bleak, Harper and Skipper proved that Thompson's group $T$ is also $\frac{3}{2}$-generated. In this paper, we prove that Thompson's group $F$ is "almost" $\frac{3}{2}$-generated in the sense that every element of $F$ whose image in the abelianization forms part of a generating pair of $\mathbb{Z}^2$ is part of a generating pair of $F$. We also prove that for every non-trivial element $f\in F$ there is an element $g\in F$ such that the subgroup $\langle f,g\rangle$ contains the derived subgroup of $F$. Moreover, if $f$ does not belong to the derived subgroup of $F$, then there is an element $g\in F$ such that $\langle f,g\rangle$ has finite index in $F$.

math.GR↗
arXiv2021

The generation problem in Thompson group $F$

We show that the generation problem in Thompson group $F$ is decidable, i.e., there is an algorithm which decides if a finite set of elements of $F$ generates the whole $F$. The algorithm makes use of the Stallings $2$-core of subgroups of $F$, which can be defined in an analogue way to the Stallings core of subgroups of a finitely generated free group. Further study of the Stallings $2$-core of subgroups of $F$ provides a solution to another algorithmic problem in $F$. Namely, given a finitely generated subgroup $H$ of $F$, it is decidable if $H$ acts transitively on the set of finite dyadic fractions $\mathcal D$. Other applications of the study include the construction of new maximal subgroups of $F$ of infinite index, among which, a maximal subgroup of infinite index which acts transitively on the set $\mathcal D$ and the construction of an elementary amenable subgroup of $F$ which is maximal in a normal subgroup of $F$.

math.GR↗
arXiv2021

On closed subgroups of the R. Thompson group $F$

We prove that Thompson's group $F$ has a subgroup $H$ such that the conjugacy problem in $H$ is undecidable and the membership problem in $H$ is easily decidable. The subgroup $H$ of $F$ is a closed subgroup of $F$. That is, every function in $F$ which is a piecewise-$H$ function belongs to $H$. Other interesting examples of closed subgroups of $F$ include Jones' subgroups $\overrightarrow{F}_n$ and Jones' $3$-colorable subgroup $\mathcal F$. By a recent result of the first author, all maximal subgroups of $F$ of infinite index are closed. In this paper we prove that if $K\leq F$ is finitely generated then the closure of $K$, i.e., the smallest closed subgroup of $F$ which contains $K$, is finitely generated. We also prove that all finitely generated closed subgroups of $F$ are undistorted in $F$. In particular, all finitely generated maximal subgroups of $F$ are undistorted in $F$.

math.GR↗
arXiv2018

Autostackability of Thompson's group $F$

The word problem for Thompson's group $F$ has a solution, but it remains unknown whether $F$ is automatic or has a finite or regular convergent (terminating and confluent) rewriting system. We show that the group $F$ admits a natural extension of these two properties, namely autostackability, and we give an explicit bounded regular convergent prefix-rewriting system for $F$.

math.GR↗
arXiv2017

On subgroups of R. Thompson's group $F$

We provide two ways to show that the R. Thompson group $F$ has maximal subgroups of infinite index which do not fix any number in the unit interval under the natural action of $F$ on $(0,1)$, thus solving a problem by D. Savchuk. The first way employs Jones' subgroup of the R. Thompson group $F$ and leads to an explicit finitely generated example. The second way employs directed 2-complexes and 2-dimensional analogs of Stallings' core graphs, and gives many implicit examples. We also show that $F$ has a decreasing sequence of finitely generated subgroups $F>H_1>H_2>...$ such that $\cap H_i=\{1\}$ and for every $i$ there exist only finitely many subgroups of $F$ containing $H_i$.

math.GR↗
arXiv2016

Invariable generation of Thompson groups

A subset $S$ of a group $G$ invariably generates $G$ if $G= \langle s^{g(s)} | s \in S\rangle$ for every choice of $g(s) \in G,s \in S$. We say that a group $G$ is invariably generated if such $S$ exists, or equivalently if $S=G$ invariably generates $G$. In this paper, we study invariable generation of Thompson groups. We show that Thompson group $F$ is invariable generated by a finite set, whereas Thompson groups $T$ and $V$ are not invariable generated.

math.GR↗
arXiv2016

Nonempty intersection of longest paths in $2K_2$-free graphs

In 1966, Gallai asked whether all longest paths in a connected graph share a common vertex. Counterexamples indicate that this is not true in general. However, Gallai's question is positive for certain well-known classes of connected graphs, such as split graphs, interval graphs, circular arc graphs, outerplanar graphs, and series-parallel graphs. A graph is $2K_2$-free if it does not contain two independent edges as an induced subgraph. In this paper, we show that in nonempty $2K_2$-free graphs, every vertex of maximum degree is common to all longest paths. Our result implies that all longest paths in a nonempty $2K_2$-free graph have a nonempty intersection. In particular, it gives a new proof for the result on split graphs, as split graphs are $2K_2$-free.

math.CO↗
arXiv2016

On the stabilizers of finite sets of numbers in the R. Thompson group $F$

We study subgroups $H_U$ of the R. Thompson group $F$ which are stabilizers of finite sets $U$ of numbers in the interval $(0,1)$. We describe the algebraic structure of $H_U$ and prove that the stabilizer $H_U$ is finitely generated if and only if $U$ consists of rational numbers. We also show that such subgroups are isomorphic surprisingly often. In particular, we prove that if finite sets $U\subset [0,1]$ and $V\subset [0,1]$ consist of rational numbers which are not finite binary fractions, and $|U|=|V|$, then the stabilizers of $U$ and $V$ are isomorphic. In fact these subgroups are conjugate inside a subgroup $\bar F<\Homeo([0,1])$ which is the completion of $F$ with respect to what we call the Hamming metric on $F$. Moreover the conjugator can be found in a certain subgroup $\F < \bar F$ which consists of possibly infinite tree-diagrams with finitely many infinite branches. We also show that the group $\F$ is non-amenable.

math.GR↗
arXiv2015

On Jones' subgroup of R. Thompson group $F$

Recently Vaughan Jones showed that the R. Thompson group $F$ encodes in a natural way all knots, and a certain subgroup $\vec F$ of $F$ encodes all oriented knots. We answer several questions of Jones about $\vec F$. In particular we prove that the subgroup $\vec F$ is generated by $x_0x_1, x_1x_2, x_2x_3$ (where $x_i, i=0,1,2,...$ are the standard generators of $F$) and is isomorphic to $F_3$, the analog of $F$ where all slopes are powers of $3$ and break points are $3$-adic rationals. We also show that $\vec F$ coincides with its commensurator. Hence the linearization of the permutational representation of $F$ on $F/\vec F$ is irreducible.

math.GR↗