Growth of quasiconvex subgroups

Dahmani, François; Futer, David; Wise, Daniel T.

doi:10.1017/s0305004118000440

Cited by 14 publications

(34 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In Section 3.3, similarly to [DFW16], we produce a left eigenvector of the transition matrix of the shortlex machine, with eigenvalue λ, whose support consists of states with "maximal growth"-that is, states whose number of n-th successors grows at the same rate as the group itself. In Section 6, we confirm that such states are dense in every shortlex shelling.…”

Section: Populated Shellingsmentioning

confidence: 99%

“…A remarkable fact about hyperbolic groups is that the language of shortlex geodesics is regular-we recall the relevant definitions here. For a detailed discussion see, for example, [ECH + 92], [CF10], and [DFW16] Definition 3.14. A finite state automaton (FSA) on alphabet S (where here S is an arbitrary finite set) is a directed graph whose edges are labeled by elements of S (for a formal definition see, for example, [GJ02]).…”

Section: Set Upmentioning

confidence: 99%

“…The set of all shortlex geodesics forms a regular language [ECH + 92, Proposition 2.5.2], called the language of shortlex geodesics in G (and with generators S.) Definition 3.16. Let λ := lim i→∞ #B(i, 1 G ) 1/i be the growth rate of G with respect to S (see for example [DFW16]). Let M denote a pruned FSA for the language of shortlex geodesics in G, and let A denote the vertex set of M.…”

Section: Set Upmentioning

confidence: 99%

See 2 more Smart Citations

Strongly aperiodic subshifts on surface groups

Cohen¹,

Goodman-Strauss²

2017

Groups Geom. Dyn.

View full text Add to dashboard Cite

This paper is devoted to proving the following theorem.Theorem. A hyperbolic group admits a strongly aperiodic subshift of finite type if and only if it has at most one end.We introduce the subject in Section 1 and give an informal outline in Section 2. In Section 3, we formally define our terms and set up the proof, which is a combination of the results of Sections 3-9 as follows:Proof of the Theorem. Propositions 8. 5, 8.12, and 9.5 show that any one-ended hyperbolic group G admits a non-empty subshift of finite type in which no configuration has an infinite order stabilizer. By Proposition 3.3, G admits a subshift of finite type in which no configuration has a stabilizer of finite order. Proposition 3.4 shows that the product of these subshifts is a strongly aperiodic subshift of finite type on G.By Proposition 3.3 every zero-ended group (that is, every finite group) admits a strongly aperiodic subshift of finite type, and [Coh17] shows no group with two or more ends admits such a subshift. 3(4): 220-234, 1960.

show abstract

Section: Populated Shellingsmentioning

confidence: 99%

Section: Set Upmentioning

confidence: 99%

Section: Set Upmentioning

confidence: 99%

See 1 more Smart Citation

Strongly aperiodic subshifts on surface groups

Cohen¹,

Goodman-Strauss²

2017

Groups Geom. Dyn.

View full text Add to dashboard Cite

show abstract

“…However, he also proved in [17] that the π 1 of a closed hyperbolic 3-manifold contains a free subgroup with critical exponent arbitrarily close to 2. Let us close this discussion by mentioning a very recent result of Dahmani, Futer and Wise [26] that for free groups F n (n ≥ 2) there exists a sequence of finitely generated subgroups with critical exponents tending to log(2n−1). Hence, an intriguing question arises concerning the conditions under which there exists a gap of critical exponents for free subgroups of ambient groups.…”

mentioning

confidence: 99%

Statistically Convex-Cocompact Actions of Groups with Contracting Elements

Yang

2018

International Mathematics Research Notices

View full text Add to dashboard Cite

This paper presents a study of the asymptotic geometry of groups with contracting elements, with emphasis on a subclass of statistically convex-cocompact (SCC) actions. The class of SCC actions includes relatively hyperbolic groups, CAT(0) groups with rank-1 elements and mapping class groups, among others. We exploit an extension lemma to prove that a group with SCC actions contains large free sub-semigroups, has purely exponential growth and contains a class of barrier-free sets with a growth-tight property. Our study produces new results and recovers existing ones for many interesting groups through a unified and elementary approach.

show abstract

“…For finitely-generated abelian groups, it is possible that the number of locally restricted compositions of a is asymptotically independent of a but it is no longer possible for each total a to be asymptotically equally likely since the group is infinite. The recent paper [20] explores other problems involving finitely-generated groups and enumeration.…”

Section: Resultsmentioning

confidence: 99%

Enumerative properties of restricted words and compositions

MacFie¹

View full text Add to dashboard Cite

Words and integer compositions are fundamental combinatorial objects. In each case, the object is a finite sequence of terms over a particular set. Relevant properties, sometimes called "parameters", are the length of the sequence and, for integer compositions, the sum of the sequence.There has been interest within enumerative combinatorics in counting words and compositions, especially restricted variations where the objects satisfy extra conditions. "Local" restrictions are related to contiguous subsequences, for example Smirnov words where adjacent letters must be different. For integer compositions or words over an ordered alphabet, a "subword pattern avoidance" restriction requires all contiguous subsequences of a fixed length to not satisfy a certain relative ordering. For example, we may count compositions not containing a strictly increasing contiguous subsequence of length three. "Global" restrictions, on the other hand, are related to arbitrary subsequences. A "subsequence pattern avoidance" restriction requires all subsequences of a fixed length to not satisfy a certain relative ordering.Beyond sequences we may also consider objects with different structures, and interpret local and global restrictions appropriately. We say "cyclically restricted" finite sequences are those where the last and first terms are considered adjacent for the purposes of the restriction, i.e. the restriction wraps around from the end to the start. "Circular" objects are the orbits of finite sequences under circular shifts, so all circular shifts of a finite sequence are considered the same object.We can generalize integer compositions by replacing the semigroup of positive integers with a different additive semigroup, giving the broader concept of a "composition over a semigroup", i.e. a finite sequence with a certain sum over the semigroup. Beyond the positive integers, we focus on semigroups which are finite groups -where such compositions are in fact also "words" in the group theory sense. Compositions over a finite group are relatively little-studied in combinatorics but turn out to be amenable to combinatorial analysis in analogy to both words and integer compositions.In this document we achieve exact and asymptotic enumeration of words, compositions over a finite group, and/or integer compositions characterized by local restrictions and, separately, subsequence pattern avoidance. We also count cyclically restricted and circular objects. This either fills gaps in the current literature by e.g. considering particular new patterns, or involves general progress, notably with locally restricted compositions over a finite group. We associate these compositions to walks on a covering graph whose structure is exploited to simplify asymptotic expressions. Specifically, we show that under certain conditions the number of locally restricted compositions of a group element is asymptotically independent of the group element. For some problems our results extend to the case of a positive number of subword pattern occurrences (instead...

show abstract

Growth of quasiconvex subgroups

Cited by 14 publications

References 36 publications

Strongly aperiodic subshifts on surface groups

Strongly aperiodic subshifts on surface groups

Statistically Convex-Cocompact Actions of Groups with Contracting Elements

Enumerative properties of restricted words and compositions

Contact Info

Product

Resources

About